What is the difference between sentiment analysis (e.g. on Facebook) and Twitter sentiment analysis?
There is nothing specifically called Twitter sentiment analysis, rather its colloquial. Sentiment analysis is a common term and could be applied on any dataset depending upon the use-case considered.

I have not came across any facebook api for scraping posts (maybe there are privacy concerns, as sometimes you cannot see fb pages when you are not an user), nevertheless there are several apis to scrap tweets from twitter, even specifying person, geographic location, language.

here is a github gist for reference, it uses python tweepy module to grab current tweets.