Question: One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is seen as the "deeper" result. 

Could someone explain this intuition? In particular, are there logical systems which are compact but not complete?

EDIT: Andreas brings up an excellent point, which is that the Completeness theorem is really a combination of two very important but almost unrelated results. First, the Compactness theorem. Second, the recursive enumerability of logical validities. Note that neither of these results depends on the details on the syntactic or axiomatic system.

What is the connection between these two aspects of completeness? Are there logical systems that have one of these properties but not the other?

Answer: 
This is a side comment. There are several answers that explain why compactness is so important in model theory, and I agree with what they say. But I want to point out that the "in model theory" part is key here. In the overall study of logic, not restricted to model theory, both compactness and completeness are important, and each of those has areas of logic that favor it. Model theory, being a semantic field, naturally identifies more with semantic notions. 

In mathematics outside logic, I think there is more implicit use of completeness than of compactness. Every time I prove that an identify is derivable from the the axioms of a group by working semantically and showing that the identity holds in every group, I am implicitly using the completeness theorem. It is easy to miss this or take it for granted, because the completeness theorem is so well known.

There are systems that do not have complete deduction systems; one example is second-order logic with full second-order semantics. In this system it is perfectly possible for something to be true in every model without being provable in our usual proof system. Therefore, when we study this system in logic, we have to keep a close watch on whether we have shown something is provable, or just shown that it is logically valid. 

Imagine the difficulties in an alternate world where mathematicians have to distinguish between "true in all groups" and "provable from the axioms of a group". The completeness theorem is what lets us ignore this. By comparison, it's more difficult to see reflections of the compactness theorem in everyday mathematics.
Compactness is a "semantic" theorem, whose statement involves no "syntactic" concepts such as proofs or provability. So it seems one should not need the latter concepts to prove compactness (and of course, one does not).