I’ve always used these two terms rather interchangeably. Saying that someone has been an inspiration, and thus has influenced my playing.
Come to think of it though, they aren’t really the same.
If someone is an inspiration, they don’t necessarily have to have musical ideas that you have borrowed. They don’t even have to be much of anything, except that they have some quality about them that, when you think about it, it drives you to make positive changes. If they do have musical ideas, then so much the easier to visualize for what we have here.
On the other hand if someone is an influence, then you may or may not find them, in and of themselves, to be inspiring. You do, however, find their musical ideas well worth pursuing. What they are saying musically (at least as you interpret what you hear they are saying) can be treated two different ways: generally or specifically.
Either this is what you are also generally thinking musically, in which case you (as the ‘influencee’, if you will) try to take what your influencer brings and further add / modify their input. Making the techniques, voicings or overall musical direction your own. Or else you envision what they say musically as being an important component in your overall expressions: then, you find the influence to be more specific so that it’s more a component in what you’re striving for, then you try to see how it fits into your overall musical ideas. Either way, you try to integrate their ideas into your own playing.