Independency: There is one Body of Christ on the earth, but the unity of the universal church is to exist only on an informal level rather than in a formal, visible form. Or, if there is sometimes some formal unity, it is voluntary among groups of congregations and there are no synods with binding authority. Formal, visible unity is required to exist, and exists with authority, only on the level of particular congregations. A member in one congregation may not, formally, be considered a member in all, and the same goes for elders. When a person is ordained to the office of elder, he is made an elder only over a particular congregation and is not part of a universal governing body (as there is no such thing). Therefore, he has no authority beyond the local congregation.
Semi-independency (or semi-congregationalism, or denominationalism): There is one Body of Christ on the earth, but the unity of the universal church is required to exist only on an informal level rather than in a formal, visible form. Some semi-independents believe that the church, in ideal circumstances, ought to exist in universal formal unity, but there is no requirement that it must always do so. It may be lawful for the universal church to have a universal governing body, but it is not required in all circumstances to have this. Congregations are encouraged to join together formally in larger bodies, with common governing councils or synods, but these bodies typically fall short of being the totality of de jure true churches. These larger "clumps" of congregations united in formal unity are usually called "denominations." It is lawful for there to be many such denominations. When a person is ordained to the office of elder, he is ordained only over a particular congregation and has no authority beyond it, except insofar as his congregation is part of a denomination with higher councils, in which case he is authorized to exercise his authority in these higher councils.
Before we close, let's look briefly at how each of these models of the church views denominational separation. In a presbyterian model, when denominations are divided, each denomination is rejecting the de jure status and authority of the other denomination(s) from which it is divided. As the Body of Christ has a mandate to exist in formal unity, and as elders have the right and duty to function collegially, the refusal to grant such formal communion to a denomination is to take a position of rejecting the de jure status of that denomination. Thus, denominational separation has an implied disciplinary character to it, as such separation implies a charge of sin, or heresy, or schism from one denomination to the others.
In the independent model, denominational separation is the normal state of the church, since there is no requirement for individual congregations to exist in formal communion with each other and no authoritative councils beyond the governing body of the local congregation. Therefore, in this model, denominational separation does not imply a rejection of the de jure status of the other churches from which one is separated.
In the semi-independent model, again, there is no requirement (at least in all circumstances) for universal, formal, visible unity in the Body of Christ, and there is no intrinsic authority or right that elders have to be part of a universal governing body. Therefore, in this model, like in pure independency, denominational division does not necessarily imply mutual rejection of de jure status and authority.
For biblical argumentation establishing the presbyterian model of the church as the correct model, see here. For an examination of how the Westminster Standards put forward a presbyterian model of the church, see here. For more, see here and in general here.
UPDATE 10/4/13: If I had to pinpoint the key, central issue dividing the presbyterian and the semi-independent models of the church, what would it be? I think it would be this: The presbyterian view holds that the formal unity of the church under a universal council of elders is an absolute moral requirement in all circumstances, whereas the semi-independent view holds that such formal unity of the church is, at best, only required in some (ideal) circumstances.
I think the Bible clearly supports the presbyterian view in its teaching that there is one Body of Christ. Scripture does not present the unity of the body as an optional thing, or as something which only applies sometimes. It is an essential feature of the church (just as it is essential to the physical body). It is essential to the nature of the church that its members exist in unity with each other in one body, and it is essential to the nature of the eldership of the church that the elders function together as a body. Therefore, true Christians inherently owe it to each other to remain in communion with each other, and elders owe it to each other to acknowledge each other as mutual parts of a universal governing body over the entire church, and so the maintaining of that recognition of each other and that unity is an absolute moral obligation that applies at all times and in all circumstances. If one church recognizes another church as a true, de jure church, it is morally obligated to choose to be in full formal communion with that church, and its elders are required to recognize the authority of the other church's elders to join together with them in a common governing body. It is never appropriate for one true de jure church to say to another, "I have no need of you," no matter what the condition of the other church is in terms of doctrine or practice. Christ has given an authority to his church to exercise discipline, and so it is permissible (and sometimes obligatory) for the church to cut off individual members, elders, or even church courts when they become corrupted to a certain degree. When this is done, the church is saying to the cut-off members, "we are putting you formally outside of the Body for the time being, until you repent." What is never acceptable is for a church to continue to accept the de jure status of another church (thus refusing to consider that church cut off from the Body) and yet to treat that church as if it were cut off from the Body by refusing to remain in full communion with it. This is to turn the one Body of Christ into many independent bodies, something that is clearly contrary to the Scriptural teaching regarding the nature of the church. This is the error of semi-independency (as well as pure independency).
UPDATE 5/26/14: Samuel Hudson, in his book entitled A Vindication of the Essence and Unity of the Church-Catholick Visible (1658), cites approvingly the London Ministers who authored Jus Divinum Regiminis Ecclesiastici (The Divine Right of Church Government), a famous work defending presbyterian church government, on the difference between presbyterianism and independency (p. 125):
[T]hey [the London Ministers] only set down the difference between the Presbyterians and Independents there [in the preface to Jus Divinum] to be in this, that the Presbyterians hold that there is one generall Church of Christ on earth, and that all particular Churches and single Congregations are but as similar parts of the whole; and the Independents (say they) hold that there is no other visible Church of Christ, but only a single Congregation, meeting in one place to partake of all Ordinances.
The distinguishing characteristic of presbyterianism (at least in contrast to independency or congregationalism) is that presbyterians hold that there is a single visible catholic church on the earth, whereas independents hold that there is not one visible church on the earth (at least in a formal sense) but that there are only particular visible churches formally independent from each other. Our modern semi-congregationalists allow for individual congregations to clump together in denominations, but, as the denominations exist independently from each other, we still have a form of independency rather than pure presbyterianism.
I highly recommend Hudson's book in general as a great defense of the presbyterian view of the universal visible church.
UPDATE 6/3/14: In an introduction to Samuel Hudson's original treatise on the church, Essence and Unity of the Church Catholic Visible, found in the Anthology of Presbyterian and Reformed Literature, vol. 5, ed. by Chris Coldwell (Dallas, TX: Naphtali Press, 1992), the writer of the introduction compares episcopalianism, independency, and presbyterianism (pp. 4-5):
Samuel Hudson (d. 1683) wrote his treatise The Essence and Unity of the Church Catholic Visible in the midst of the "grand debate" in the Westminster Assembly over the government of Christ's church. The Assembly was divided into three factions, as was the nation as a whole.
The first faction, that which might be called the episcopalian, took the view of church government that the bishops had served the church well in the past and could possibly do so again in the future. Some, though certainly not all, maintained that episcopacy is the form of government taught in Scripture. The majority of this faction, however, admitted that the best argument for episcopal church government was its long-standing use in the church.
The second faction, that of congregationalism or independency, took the view that each particular church contains everything needed for the church within itself. While relations with other churches are possible and at times even preferable, there is nothing in Scripture which demands a connection with other churches, according to this view.
The third faction, that which carried the day at the Westminster Assembly and to which Samuel Hudson belonged, is that of presbytery. By the time Samuel Hudson wrote, presbyterians had developed a view of jus divinum or "divine right" of church government. By this, they meant that not only is connectionalism allowed and even preferred, it is commanded by implication in Scripture. In their view, the congregationalists had virtually denied the catholicity of the visible church.
Presbyterianism holds that the entire catholic church visible is to function as one universal visible body, while congregationalism holds that it need not do so, but it is permissible for there to be multiple, independent, legitimate factions within the Christian world. The presbyterians accuse congregationalism of in essence denying the existence of the catholic church visible because of this. Since the semi-congregationalists deny the requirement for the whole church to be united in one body, this puts them in principle in the camp of the independents rather than the presbyterians, even though they hold that it is good (and maybe even required) for individual congregations to unite in more-or-less larger "clumps" of churches called "denominations."
UPDATE 6/9/14: Here is a well-articulated comment from church historian James Walker:
The visible church, in the idea of the Scottish theologians, is catholic. You have not an indefinite number of Parochial, or Congregational, or National churches, constituting, as it were, so many ecclesiastical individualities, but one great spiritual republic, of which these various organizations form a part. The visible church is not a genus, so to speak, with so many species under it. It is thus you may think of the State, but the visible church is a totum integrale, it is an empire. The churches of the various nationalities constitute the provinces of this empire; and though they are so far independent of each other, yet they are so one, that membership in one is membership in all, and separation from one is separation from all . . . This conception of the church, of which, in at least some aspects, we have practically so much lost sight, had a firm hold of the Scottish theologians of the seventeenth century.' (James Walker, The Theology and Theologians of Scotland. Edinburgh: Knox Press,  1982. Lecture iv. pp.95-6.)