The TU-160 is like a B-1A on steroids. Huge payload, very high maximum speed, very long range. That being said, I don't think it has much low level ability. This might change with the modernization program. I don't think the current models have terrain following radar.
The B-1B is optimized for low level flying (think it can do Mach 1.2 on the deck). A study in the late 70s showed that very few or any U.S. aircraft would have been able to penetrate Soviet airspace. This is why the B-1A was cancelled. Also, the B-2 was secretly being produced at the time.