The short answer is that $(\infty,1)$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn't remember all the information and hence is not always applicable.

When you work with $(\infty,1)$-categories, you have to deal explicitly with higher-dimensional coherence all the time. Everything is determined only up to equivalence. This can be kind of a pain, so it's convenient to have 1-categorical structures that neverthless carry $\infty$-categorical information.

The classical kind of 1-categorical structure used for this is a Quillen model category. This carries "too much" information, in that two objects can be equivalent in an $(\infty,1)$-category but not isomorphic in the model category it presents, and similarly two model categories can present equivalent $(\infty,1)$-categories but not be equivalent categories. Thus a model category needs a notion of "weak equivalence" between its objects, and similarly we have a notion of "Quillen equivalence" between model categories. But a model category contains all the information of an $(\infty,1)$-category, and so we can work with all the higher coherences as necessary.

A derivator is sort of a "dual" to a model category: it carries "too little" information. An advantage is that it is not subject to the issues of weak equivalence: two objects of an $(\infty,1)$-category are equivalent if and only if they are isomorphic in the corresponding derivator, and similarly two (locally presentable) $(\infty,1)$-categories are equivalent if and only if their corresponding derivators are equivalent (in a 1-categorical sense: derivators are the objects of a 2-category just like 1-categories are, and here we mean equivalence internal to that 2-category).

But a derivator doesn't have enough information to do everything we might want to with higher coherences. It's essentially an enhancement of the homotopy 1-category (the quotient by homotopies) that remembers the notion of homotopy-coherent diagram, and therefore also of homotopy limit and colimit (and homotopy Kan extension). This is sufficient for a surprising amount of homotopy coherence, at least when working only within a single $(\infty,1)$-category. But there are some things it can't do (or not very well), notably those that involve diagrams *of* $(\infty,1)$-categories and things like functor $(\infty,1)$-categories.

So the answer to your first question is yes, there are things you can do with $(\infty,1)$-categories but not with derivators; but no, anything you can do with a derivator can also be done with an $(\infty,1)$-category. And I think this mostly answers your second question as well: given that derivators are not good enough for everything, even if they make certain things easier, it's understandable that many people prefer not to learn two different languages and stick to $(\infty,1)$-categories even if they happen to be doing something that might be a bit easier with derivators. (On the other hand, I personally feel that it is easier to be sloppy with $(\infty,1)$-categories --- partly because being precise is so much work --- and easier to be precise with derivators, which is one reason that I still use the latter sometimes.)

Historically, derivators were introduced (by Heller, Grothendieck, and Franke, fairly independently) before practical notions of $(\infty,1)$-category were available. So at the time they were the only way of getting rid of the "weak equivalences"; but even with that advantage they never really caught on. I'm not quite sure why not; perhaps the requisite 2-category theory was also offputting at the time?

As for references, I don't have any suggestions other than those on the nLab page. I particularly like the expository aspect of Moritz Groth's work. You may also be interested in this blog post of mine.

sheavesof derivators? One of the great strenghts of ∞-cats is that they work very well in families (so, for example, you can rephrase faithfully flat descent as "$\mathrm{QCoh}(-)$ is a sheaf") $\endgroup$ – Denis Nardin Jul 27 at 17:46