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: 
I have two, somewhat conflicting, views on this. 

From the point of view of modern model theory, the Compactness Theorem is central to almost everything in the model theory of first order logic, while the Completeness Theorem is almost irrelevant. Completeness comes into play when proving the decidability of complete recursively axiomatized theories or in dealing with recursively saturated models for example, but these are not very central. Even when we want to apply Completeness, what's important is that the set of logical consequences of $T$ is recursively enumerable in $T$--the details of the proof system are of no importance whatsoever. This explains why in my model theory book, while I do explain that Compactness is an almost trivial consequence of Completeness, I give a direct Henkin-style proof.

Nevertheless, I view the Completeness Theorem as one of the great intellectual achievements of our subject. The fact that the semantic notion of "logical consequence" can be captured
by the syntactic notion of "proof" is really surprising. The first is, a priori, $\Pi\_1$ over the universe of sets, while the second is recursively enumerable. I find this amazing.
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).