DO NOT, under ANY circumstances, take the train. It is unreliable, especially this time of year, and could arrive late. I'm speaking from experience. Rent a car and park at a private lot north or south of the stadium at least two hours before kickoff, tailgate or find a bar/restaurant in Patriot Place to hang out.
I took the train this year for the Browns game and it was just fine.
Yes, it's been known to arrive late, but that seems rare at the moment. Admittedly, the Browns was a 1 p.m. game and the stadium was far from packed but we were there in time to drop $14 a cup on Bud Light and as much again on a sandwich. The train is very slow, and it's hard to get a seat on the way back even if you go directly to the train after the game ends (which you must). But it's pretty well marshalled and people are friendly (the more so if the Pats win!). The new thing compared with previous years is that you can (have to) buy your ticket on-line and put it on the phone.
Bottom line. The train is fine if you know what to expect. If you want to stay in Boston and enjoy the city, take it by all means, but don't think to combine that with Patriots Place or tailgating.