[Edition. In this answer I wrote about "philosophical logic" and "mathematical logic" in usual sense of these terms. But I also think that the difference between them is very conventional. For example, people studying knowledge formally by means of modal epistemic logics (as formal models) and those who studying provability logics to a large extent (if one forgets about actual motivations) are doing the same things -- they are investigating various modal logics.]
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"My question is about relations between logic as part of philosophy and mathematical logic from the second half of the 20th century when its seems that connections between these two areas have weakened." "I am quite curious also about the reasons for the much weaker connections between mathematical logic and formal models developed by philosophers at the later part of the 20th century." The main reason is that mathematical logic concerns with models of mathematical thought, but philosophical logic builds models for various parts of philosophy which are very different from mathematics (i. e. they may use modalities, analogy, induction and so on). To give one example of "formal models developed in philosophy that had become important in mathematical logic", Provability logic is such an example. To quote from Stanford Encyclopedia of philosophy: "Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates." So, provability logics are modal logics which capture provability in formal systems of mathematical logic. Provability logic is regarded as an area of mathematical logic (which represented by Robert M. Solovay, George Boolos, Sergei Artemov, Lev Beklemishev, Giorgi Japaridze, Dick de Jongh and so on) and it uses models, ideas and techniques from modal logic which is (in any case, was) part of philosophical logic. "Another thing I am curious about is to what extent for the applications to computer science formal models described by philosophers turned out to be useful." Again, the original papers by Kripke was by no means concerned with computer science. Kripke discovered another way of interpreting the modalities and the context was philosophical one. But today the whole field of program verification in CS is based on modal logic. A lot of specific modal logics are developed in CS which model specific aspects (temporal logics, dynamic logics and so on). Crucial results about them such as expresiveness, decidability and completeness are proved using essentialy the same ideas as those from original papers in philosophical context. Finally, to speak "about works in philosophical logic that were motivated or influenced by developments in mathematical logic", I quote from what I said in the meta thread: "A new interesting field in philosophy is growing in this century often called Formal Philosophy (this term I believe has its origin from the title of Richard Montague's collected papers book). People of this field (among them R. Montague, H. Putnam, E. Zalta, D. Bonnay...) are trying to solve philosophical problems using formal logic. Particularly, and this is important, many of them try to formalize such kinds of reasoning as modality, induction, analogy, simplicity, naturality, generalization and so on as well as concrete philosophical theories (which are of course in the domain of philosophical logic) using formalisms and methods of mathematical logic.
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