What is the connection between the criticisms offered by Wittgenstein and Quine of meaning and language? Are both philosophers generally criticizing the same semantic theories with similar arguments, and if not, are there any cases of genuine 'overlap' between the two?
Yes and no. They both criticize a certain approach to semantic theory that can be called realism about meaning. Roughly, realists see meanings as some kind of entities, although there is a wide range of opinions as to their nature. For Plato and Frege they are ideal forms occupying a separate realm, for Aristotle and Russell they are invariances of sensible experience, and in informational semantics of Fodor and Dretske they are correlations between the brain and external reality. Opposed to it is pragmatism about meaning advanced by Quine and Wittgenstein, which ultimately reduces meaning to assertions and speech acts, in other words semantic categories like meaning and truth supervene on "use", i.e. linguistic practice. This has purely semantic consequences. The first is holism, concepts only acquire significance as parts of a conceptual scheme, not on their own. In Fodor's stark contrast a being may possess a single meaningful concept according to realists, but not pragmatists. The second is access to meaning, according to pragmatists there is nothing to it beyond what is publicly accessible through "overt behavior in overt circumstances" as Quine put it, this is seen as essential to it being learnable and communicable. Finally, without meanings as entities truth and reference can not be viewed as correspondence relations, their role is deflated to being phrasing devices, e.g. "p is true" is a figure of speech for asserting p indirectly.
There is however a second dimension to the spectrum of semantic theories, on which Quine and Wittgenstein are at the opposite ends. Classical semantics starting with Mill and Frege was truth-conditional, meaning of sentences was seen as reducible to conditions of their obtaining. This entails for example that like classical truth it is compositional, meaning of complex sentences reduces to meanings of their constituents. Moreover, the account of inference for complex sentences can be given based on truth tables in terms of truth of elementary sentences. Quine never offered a semantic theory of his own, but his star pupil and heir apparent Davidson did offer one that is both pragmatist and truth-conditional. The derivative and relative nature of meaning in it manifests in the fact that multiple global assignments of truth-conditions may work equally well, even as they disagree on which sentences are true. This is the famous indeterminacy of translation/interpretaion.
Wittgenstein was led away from truth-conditionalism after the Tractatus when he encountered the so-called "color problem". Both "A is red" and "A is green" seem like elementary sentences, yet the first implies the negation of the second. Truth-conditional semantics has difficulties accounting for such "material inference", its notion of inference is a formal one. So (intermediate) Wittgenstein came to view meaning as derivative from the web of inferences that a sentence participates in, its inferential role, and language as a web of meaning-giving overlapping "calculi". This approach came to be called inferentialism. Inference is treated as a primitive, and truth is seen as not only deflated as in all pragmatist accounts, but a defective notion. As semantic paradoxes like the Liar show it can not be coherently reconstructed from basic notions without limitations. A big event in 1990s was Brandom giving a first full fledged account of inferential semantics for natural languages in Making It Explicit (he credits Dummett and Sellars as key links in transmission from Wittgenstein).
One could say that Quine and Wittgenstein agree on language philosophically, and their disagreement is more of a technical one. Ultimately, it stems from difference in purpose: Quine was more concerned with accommodating scientific practice and scientific language, and sceptical about the prospects of a semantic theory for natural languages. Wittgenstein, on the other hand, was always more interested in natural language, even already in the Tractatus. The irony is that in his late period he came to where Quine started, concluding that nuances of context and application make any semantic theory of natural language hopeless. His late philosophy, of language in particular, is not to build theories but to therapeutically nudge others to realize the confusions "bewitchment by means of language" leads to. A good review is MacFarlane's Pragmatism and Inferentialism, see also Johnsson's Quine and Kripke's Wittgenstein.