I think one major difference is that MPs are far less beholden to their party for reelection and it is not uncommon for them to cross the floor when they feel the interests of their constituents are not being represented by the governing (or opposing) party.

Yes, a PM with a whipped majority is tremendously powerful, but getting that whipped majority is not an easy task and requires significant politicking and negotiating within the party precisely because individual MPs are proportionately more powerful than legislators in the US.