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.
I was to say that the answers given so far are all wrong and misleading, but thanksfully I recalled that I am not a mathematician :-)

There are mainly two approaches to the concept of "logic".

1. The classical (or mathematical) approach to logic. Roughly speaking, a logic consists of two classes: a set of formulae and a class of models, together with a satisfiability relation saying what formulae are true in what models. Then, we may develop a proof system (or various proof systems) for the logic, which helps us --- in a systematic and coherent way --- derive satisfiability of formulae. Desirable properties of such proof systems are soundness (what we have derived is true), completness (what is true, we can derive). There is also compactness (if something follows a theory then it follows from a finite subset of the theory) that refers to the logic itself (here: satisfiability relation). This is how mathematicians are taught logic.
2. Modern (or computer-scientistic) approach to logic. Logic is a kind of a formal system (deductive system). To help proving facts about such a system, we may introduce the concept of "models" (or various concepts of models) for the logic. Desirable properties of such classes of models are that the deductive system over them is sound and complete (that is --- for a given system --- we develop the appropriate concept of models, such that the proof system is sound and complete; if we add/remove some axioms/rules to the system then we have to restrict/extend our class of models; this is most easily seen in temporal logics --- for example, LTL is sound and complete in *linear* models). There is also compactness (if something follows a theory then it follows from a finite subset of the theory) that refers to the logic itself (here: proof system; if a logic allows only finitary proofs, then it is obviously compact). Simply, in this approach, the system is fundamental here. This is how computer scientists are taught logic.

Of course, in the presence of soundness and completeness, classical compactness and modern compactness coincide.

So, moving back to your question --- I do agree with other answers saying that completeness and compactness are just far different concepts, so neither is "deeper". However, I do not think that the classification of wht belongs to models and what belongs to proofs is that obvious --- it is just all about how you think of logic.