Tennessee had a bigger need at OLB than the Pats did, and therefore were going to be more willing to pay Woodyard more money. Zach Brown was benched last year, and he plays the position best suited for Woodyard, WLB. Brown is still on a rookie contract, so they can easily afford to keep him on the roster and see if he still develops, or at least keep him for depth. The Titans may have also had a bit of an inside edge in that Woodyard's college DC is on the Tennessee staff, and he is closer to home (grew up in Georgia, played at Kentucky in college).
The Titans still have a major hole to fill at MLB; for their sake I hope they don't intend on using Woodyard there. Last year showed that he is much better suited to playing outside than inside.
Bummer that the Pats were unable to sign Woodyard, but I can understand why that happened. The next best free agent OLBs available are a step down (Rob Jackson, Parys Haralson), but will also cost less.
Pats also need an ILB; Pat Angerer is still on the market, is only 27, and should be better than last year when his knee injury limited his productivity. Angerer was a tackling machine his first two years, but missed a lot of time the last two. If he can pass a physical then he could be a decent signing.