I think that everything important that can be said about the differences between Compactness and Completeness Theorems and their proofs from the technical point of view has been said. (I also like most the detailed and elucidating answer given by Joel David Hamkins (at http://mathoverflow.net/questions/9309/in-model-theory-does-compactness-easily-imply-completeness)). On the other hand, one of the most important differences between these theorems is a non-technical one, and indeed some previous answers contain hints to this effect. Indeed, Completeness Theorem has an obvious metamathematical (or even philosophical) flavour as opposed to Compactness Theorem. Actually, it is about the relation between the two most important mathematical notions, i.e., those of proof and truth.
And here I would like to argue with those (Carl Mummert and Stefan Geschke) who claim that sometimes Completeness Theorem is used in everyday mathematics. Actually, as I see it, it is about everyday mathematics, but it does not belong to everyday mathematics.
Actually, contrary to what Carl Mummert says, I doubt that, in everyday mathematics, anybody in any time uses completeness theorem in either an explicit or implicit way. Obviously, one can successfully work in any field of mathematics (which are not intimately connected to logic) without any knowledge of mathematical logic. (Clearly she or he has to have a good sense of logic, but this is a completely different matter.) In other words (unlike Carl Mummert), I cannot imagine any `difficulties in an alternate world where mathematicians have to distinguish between "true in all groups" and "provable from the axioms of a group" '. The reason is simple. I do not think that anyone proves "that a group identity is derivable from the axioms of a group by working semantically and showing that the identity holds in every group." Though I am not a group theorist, I think that no group theorist is interested in the statements that are provable from the axioms of group theory alone. (On the other hand, of course, the most important elementary statements needed to begin group theory at all are usually derived directly from the axioms.) Most mathematicians work in intuitive set theory and freely make use of the different possibilities that this rich theory offers (independently of the fact that she or he is aware of the existence of ZFC). (Actually, the notion of a group itself is defined as a model, that is, generally in terms of sets rather than a first order theory. And, of course, this kind of definition is very practical, since otherwise every course on groups have to be preceded by an introduction to logic.) I think that the pure first order theory of groups has only theoretical or didactic significance for being a nice widely known example of a first order theory.
Likewise, I do not agree with Stefan Geschke that "the completeness theorem does explain why we can do mathematics the way we do." Just the other way around. Clearly, metamathematics is the study of real mathematics by exact mathematical means. Therefore, its notions are intended to mimic those of everyday informal mathematics as faithfully as this is possible. So a metamathematical result cannot explain or justify anything. What it can do is to describe in exact terms and clarify the way mathematics is normally done (and, of course, to draw consequences about everyday mathematics from the results of this description). But its results do not affect the way mathematics is normally done. Obviously, we would do everyday mathematics in exactly the same way if the Completeness Theorem did not hold. Just as those mathematicians do who never have heard of this theorem. And indeed, we do arithmetic in exactly the same way as mathematicians before Gödel (who might well think that true arithmetic was completerecursively axiomatizable) did.