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.
Here is an interesting quote from Bruno Poizat's "A Course in Model Theory":

> 
> The compactness theorem, in the forms of Theorems 4.5 and 4.6, is due to Gödel; in fact, as explained in the beginning of Section 4.3 [Henkin's Method], the theorem was for Gödel a simple corollary (we could even say an unexpected corollary, a rather strange remark!) of his "completeness theorem" of logic, in which he showed that a finite system of rules of inference is sufficient to express the notion of consequence (see Chapter 7). It could also have been taken from [Herbrand 1928] or [Gentzen 1934], in which results of the same sort were proven.
> 
> 
> This unfortunate compactness theorem was brought in by the back door, and we might say that its original modesty still does it wrong in logic textbooks. In my opinion it is a much more essential and primordial (and thus also less sophisticated) than Gödel's completeness theorem, which states that we can formalize deduction in a certain arithmetic way; it is an error in method to deduce it from the latter.
> 
> 
> If we do it this way, it is by a very blind fidelity to the historic conditions that witnessed its birth. The weight of this tradition is apparent even in a work like [Chang-Keisler 1973], which was considered a bible of model theory in the 1970s; it begins with syntactic developments that have nothing to do with anything in the succeeding chapters. This approach---deducing Compactness from the possibility of axiomatizing the notion of deduction---once applied to the propositional calculus gives the strangest proof on record of the compactness of $2^\omega$!
> 
> 
> It is undoubtedly more "logical," but it is inconvenient, to require the student to absorb a system of formal deduction, ultimately quite arbitrary, which can be justified only much later when we can show that it indeed represents the notion of semantic consequence. We should not lose sight of the fact that the formalisms have no raison d'être except insofar as they are adequate for representing notions of substance.
> 
> 
> 

See also this [old answer of mine](https://mathoverflow.net/questions/9309/in-model-theory-does-compactness-easily-imply-completeness/11014#11014) where the same quote appears.