The link between philosophy and science is not so straightforward, it is rarely possible to say that some scientific developments are directly responsible for some specific philosophical developments. This said, one general trend started even before the term "neuroscience" became fashionable, and "neurophysiology" was more common. Burge describes it in Philosophy of Language and Mind: 1950-1990:
"During the mid to late 1960s materialism became one of the few orthodoxies in American philosophy. It is difficult to say why this happened. No single argument obtained widespread acceptance. Perhaps the success in biochemistry during the 1950s in providing some sense of the chemical underpinnings of biological facts encouraged the expectation that eventually mental facts would receive a similar explication in neural terms. Moreover, there were some spectacular advances in animal neurophysiology during the period."
This tendency was solidified by the emerging human neuroscience as the prospect of finding neural correlates of mental states drew closer, and led to the current dominance of physicalism in the (analytic) philosophy of mind. Books like Dennett's Consciousness Explained and Churchland's Matter and Consciousness became fashionable among general public and even scientists (Crick in The Astonishing Hypothesis explicitly refers readers to Churchlands for philosophical sustenance).
Indirectly, neuroscience stimulated interest of philosophers in developing semantic models that parallel brain function. Although much of constructive work is based on speculative analysis and artificial neuro-nets, results of neuroscience experiments are thoroughly monitored, interpreted, and incorporated as far as possible (and perhaps further given the relative crudeness of experimental setups at this point). There is active work on such models of concept acquisition, see e.g. Mandler's On the Birth and Growth of Concepts, creative cognition, see Thagard on neural mechanisms of abductive inference, and neurosemantics, see Eliasmith.
The outsized impact of the Libet experiment is due to the fact that it seemed like one place where experiments had a direct bearing on a genuine philosophy of mind problem, the problem of voluntary control due to the link between the so-called readiness potential (RP) and conscious intentions. This link appears doubtful today, but it forced many philosophers of volition with libertarian leanings to seriously consider the possibility of unconscious free will as an alternative to Libet's "conscious veto". Roskies in How Does Neuroscience Affect Our Conception of Volition? writes:
"Moreover, some lines of evidence from psychology and neuroscience suggest that actions are often initiated unconsciously (e.g., Libet 1985), and so if free will is to exist, it will take the form of control or veto power over unconsciously initiated actions. Finally, regardless of whether one believes that we can act or choose freely, we normally do perceive certain actions as self-caused and others as not. Most people concur that there is phenomenology that accompanies voluntary action, and neuroscience has begun to illuminate the physiological basis of the feeling of agency."
Neo-Wittgensteinians, like Hacker, even see unconscious will as a vindication of Ryle's knowledge how and a way out of Wittgestein's rule-following paradox, see What counters are there to Spinoza's argument that acts of free will create infinite regress?:
"When one utters a sentence, every word is spoken voluntarily, but it would be ridiculous to claim that one consciously performs successive acts of will, one for each word (or phoneme?) an instant before utterance... The willing must not be conceived as doing something, the doing of which then causes the movement of one’s body. That would be a case of bringing about the movement of one’s body by doing something else. Rather, the willing would have to be an ‘immediate causing’".
Of course, the old school hard determinism also got boosted, this time under the new nickname of "willusionism", forcefully promoted in Wegner's The Illusion of Conscious Will, see What philosophers, other than Schopenhauer and Nietzsche, examine "will" first, before "free will"?:
"The initiation of the voluntary act appears to be an unconscious cerebral process. Clearly, free will or free choice of whether to act now could not be the initiating agent, contrary to one widely held view. This is of course also contrary to each individual’s own introspective feeling that he/she consciously initiates such voluntary acts".
In a way, history repeats itself. Similar rise of (then) mechanistic materialism and determinism, fueled by the rise of physics and "psychophysiology" of the day, occured at the end of 19th century, and was then championed by some prominent scientists, like Bernard and Du Bois-Reymond, see History of the study of indeterminism in classical mechanics. Measurable experimental connections established between sensations and stimuli, such as the Weber-Fechner law, led to unrealistic expectations about building a physical theory of the mental, and soon. Experiments, however, do not create theories on their own, although they can inspire them.
What makes me more hopeful this time is the potential confluence between neuroscience, AI research, especially embodied cognition a la Dreyfus and brain models based on deeper understanding of (artificial) neuro-net operation (see e.g. Thagard's How Brains Make Mental Models); and instrumentalist semantics (which promises an advance on the problem of intentionality, relating the mind to its referents, see Can there be a sufficient account of meaning without an account of intentionality?).