Isn't there a cheat sheet that someone put together that highlights the major changes from BMR to TW?
I'm about to loan my BMR to a friend so he can spool up.
If you are only worrying about mechs then there are few changes. Vehicles and Infantry saw the biggest changes.
#1) Partial cover now is a +1, full hit table, legs hits are ignored (hit the hill)
#2) Some critical hits are slightly different (Hip is the main one)
#3) AMS now only spends 1 ammo per use, and is a flat -4 against the missile roll (instead of rolling for it)
If facing infantry, it is a major change, since big anti-mech weapons do almost nothing to them, but flamers and machine guns slay them.
Other than that stuff, it is pretty close to the same. Enough that you can play. You will still want to look at the TW errata (linked from these forums) as it has some clarifications, or just pick up the Battlemech Manual.