I gave at talk about this at Scala Days NY, and I'll give it again in Berlin in a few weeks if I remember to actually book transport and a hotel. I'll just try and summarise here, because I don't have the energy or time to write out my talk :-)

I think FP and OO are fundamentally different in how they approach programming: the way they break down problems, what they consider to be good code, and so on. OO views code as small unit of encapsulated state communicating with one another. FP views code as a series of transformations. OO code is not good FP code, and vice versa.

On a lower level you can find many similarities. OO patterns are largely FP language features, from a time when knowledge of FP was not very widespread. At least five of the Gang of Four patterns are first class functions is disguise. Going the other way, a very typical pattern in functional programming is to build up a description of what should happen, and then have an "interpreter" actually execute that description. These interpreters often look like "coalgebraic" structures, which is an FP term for ... objects. Now isn't that interesting?

In summary, the techniques often look very similar if you squint, but the fundamental way they approach programming is very different.

That's a very quick sketch, which is somewhat hazy on the details. Apologies for the brevity and lack of nuance.