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: 
The point is that we care about the models, rather than about the proofs. The compactness theorem---the claim that a theory is satisfiable iff every finite subset of it is satisfiable---is fundamentally connected to the models, and the possiblility of truth in these models. To use it, you need to understand your theory, the models of your theory and the models of finite pieces of your theory. And this is what we want to be thinking about and what we know about. In particular, when working with models, we can use all the mathematical tools and constructions at our disposal, with no need to remain inside any first-order language or formal system (well, perhaps we have set theory as our background system). We are free to reason about the models via reducts and ultrapowers and limits of systems of morphisms and so on, using any mathematical method at all.

The completeness theorem, in contrast, is fundamentally connected with the details of a formal deduction system. And so when using it, one is thinking about whether certain tautologies might be provable or not, or whether a certain formal consequence is allowed in the system or not. 
But when we are studying a certain first-order class of groups or rings or whatever, such details about the proof system might seem to be an irrelevant distraction.
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.