How come I can have separate a/cs going on the Twitter mobile app (quick switch between each), but have to sign off then back in to another, if using web version..?
Because Twitter want to keep the website simple for new users. They have a problem with people trying it, not really understanding it, not having anyone to tweet to, and never coming back.
You might want to try TweetDeck, which was one of the most popular third-party apps before Twitter acquired it, but still seems to provide a kind of power user web interface. Also available as a Chrome app or standalone application for Windows/Mac. I don't use it myself but I've heard good things.