Pats offered basically the same as seattle, think it was around 6 mill a year,he did get more but nothing substantial (imo when we're talking 6 mill), one main difference was the pats werent willing to rip up this years contract to start it. since contracts arent guaranteed it doesnt matter if they offer him 3 years or 20 years anyone can cut him after 1.
From the patriots perspective they dont want everyone who has a good year to hold out for more money when they have one or more years left on their contract as branch did.
Just because someone is willing to pay more doesnt mean the highest price was "fair" value, lots of these teams make silly financial moves on "superstars". Pats have been good about it, we've lost some greats doing it but 3 superbowls in 5 years if pretty good. I'm a season tix holder and would have loved to see branch out there.