The primary difference between Catholic and Protestant theology on the issue of concupiscence is as follows: Protestants consider concupiscence to be sinful, whereas Catholics believe it not to be sinful in itself, although it is highly likely to cause sin.
This difference is intimately tied up with the different traditions differing views on original sin. Catholicism teaches that humanity's original nature contained an innate tendency to sin, but due to a special supernatural gift granted by God to Adam and Eve, namely original righteousness, they were able to overcome this innate tendency, and fully orient themselves towards God. However, due to the Fall, they lost this special gift. As a result, their natural self rules, which is not fully oriented towards God, and this lack of orientation results in sin. However, despite its resulting in sin, since it is natural it cannot be called evil, but in itself must be recognized as good; and in and of itself it is not the cause of sin, although once it comes into contact with sin it proceeds to produce a multitude of more sin -- if sin is a fire, the nature of humans is like petrol (or gasoline to Americans) -- by itself it is not alight (i.e. not sinful), but when it comes into contact with the tiniest bit of fire (i.e. of sin), it bursts into flames. This original nature of human beings is called concupiscence.
Protestantism, by contrast, holds that the original prelapsarian nature of humanity was an innate tendency to good; the special relationship Adam and Eve enjoyed with God was due not to some supernatural gift, but to their own natures. Hence the Fall was not the destruction of a supernatural gift, leaving humanity's nature to work unimpeded, but rather the corruption of that nature itself. Since human's present nature is not their original nature, but rather corrupted, it follows that it is not good, but rather evil (although some good it may still contain.) Thus, in the Protestant view, concupiscence, rather than being merely a good which reacts rather badly to the presence of even the smallest evil, is evil in itself.
Another source for the differing views of Protestants and Catholics on concupiscence is their differing views on sin. Protestants (or at least the magisterial reformers; some modern day Protestants would doubtless not accept this position) hold that one can be guilty of sin even if it is not voluntary; Catholics by contrast traditionally hold that one is only guilty of sin if one's sin is voluntary. It is to be noted that the Scholastics and the magisterial reformers have rather different views on the issue of what is voluntary and what is not -- the Scholastics considered the emotions of love, hate, like and dislike that one experiences to be acts of will, while the reformers did not consider them to be as such. Since, according to the Bible, it is not just one's actions which are sinful, but also one's attitudes, some of these attitudes must be sinful. If one accepts the Catholic position that one's attitudes are acts of will, then one can still maintain that all sin is voluntary; but if you accept the position of the magisterial reformers, that these attitudes are involuntary, it must therefore follow some sins are involuntary. Since man's nature is clearly not voluntarily chosen by him, by the Catholic position it must not be sinful (since all sins are voluntary); but by the reformer's position, since some sins are involuntary, it can be.
The difference in views also extends to the relationship between concupiscence and original sin. In the Protestant view, original sin is concupiscence inherited from Adam and Eve. It is never fully eliminated in this life, although sanctifying grace helps to eliminate it gradually. Since in the Catholic view however, concupiscence is not sinful, it cannot be original sin. Rather, original sin is the real and actual sin of Adam, imputed on to his descendants; but unlike the Protestant view, rather than necessarily remaining until death (or in the case of the damned, for all eternity), it can be removed by the sacrament of baptism. The implications of this should be discussed in the article on original sin.
As mentioned in the introduction, concupiscence, although known in Protestantism, is primarily a Catholic term. This is because Protestants believe concupiscence to be sinful, indeed, they believe it to be the primary type of sin; thus they most often refer to it simply as sin, or if they wish to distinguish it from particular sinful acts, as "man's sinful nature". However, since Catholics reject this as being a sin, they must find another term to refer to it as; thus they refer to it as "concupiscence".
References.
Robert Merrihew Adams, "Original Sin: A Study in the Interaction of Philosophy and Theology", p. 80ff in Francis J. Ambrosio (ed.), The Question of Christian Philosophy Today, Fordham University Press (New York: 1999), Perspectives in Continental Philosophy no. 9.
NOTE: This article really needs review by a theologically educated Catholic, and a theologically educated Protestant. It could also do with a broader range of sources, some concrete examples, and information on the views of the Eastern Orthodox.