I've always loved the series. Whilst from time to time the micro stuff - the cigars held by the men - do jump out at you and say 'woah! so 50's' the story itself has held up well over time.
Spoilers for Foundation and Earth
Parallels can be drawn between the Shivans and the threat of extragalatic invasion that made Golan Trevize ultimately go for Gaia. Both are alien and cannot be neogotiated with, both are powerful, both have the ability to divide the current population (HoL and later Bosch), and both cannot be handled by the socio-prediction systems because they are alien
There are fruther parallels. With soft power the leaders at top are not democractically elected by the populace, at least not in regular elections. They are academically accomplished and they need to be. With the hard powers it's more difficult to say. The BP GTVA seems to have democractic insitutions but we don't how it works. The Foundation changes from time to time.
As I said in another thread, psychohistory is in some ways merely the pinnacle of what we have today in large-scale social sciences. That there would be some parallels is not a surprise. But Foundation, as Battutan noted, is not concerned merely with psychohistory. It is also, amongst other things, concerned with the larger themes of power and the individual versus the historically inevitable. And... well... I'll leave it there for now.