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: 
Really, compactness is seen as the deeper result? I have to say that I am also more interested in the models rather than formulas and deduction systems, and hence like
to teach students proofs of the compactness theorem that do not use the completeness
theorem. Typically I prove compactness using ultraproducts. 

On the other hand, I still believe that the completeness theorem for first order logic
is the most important theorem of mathematical logic.  

The theorem tells you that in principle there are computer checkable proofs for all "true"
theorems. I know that many mathematicians are not really concerned with having
a solid foundation for the concept of a "proof" (you know it when you see it).
But having this formal concept of proof in the background helps tremendously when 
you want to fight off people who present their $n$-th proof of the inconsistency
of PA or ZFC, or that there are no infinite sets or that CH holds.
Also, the completeness theorem does explain why we can do mathematics the way we do.
Even though nobody really writes formal proofs of anything, ever, unless the correctness
of the proofs is seriously challenged (see the FLYSPECK project at <http://code.google.com/p/flyspeck/>).

Also, I think that the proof of the completeness theorem is deeper than of the compactness
theorem. In some sense the proof of the completeness theorem (I am thinking of the proof where one builds a canonical model of a maximally consistent Henkin theory) is more straight forward than for example the ultraproduct proof of the compactness theorem,
but it is more complicated in the details and certainly less accessible to mainstream mathematics than for instance the ultraproduct proof of the compactness theorem.
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.