The new Tu-160M2 will probably be used for the same kind of missions as the older Tu-160M version, and carry the same Kh-101 and nuclear Kh-102 cruise missiles.
Source: The Drive
Including improved inertial, satellite, and astro-navigation equipment. An active electronically scanned array radar will be to able to detect targets at longer ranges and with better accuracy.
Source: The Drive
It will also reportedly have new avionics, sensors, and communications equipment.
Source: The Drive, The Aviationist
It will reportedly have a new engine — four of the NK-32 02 series seen below — which will add 600 miles to its flying range.
The original engine on the Tu-160 — the NK-32 — is already a powerful engine, providing 55,000 lbs of thrust. The B-1B Lancer, on the other hand, only generates about 31,000 pounds of thrust.
Source: The Drive, The Aviationist
It's full capabilities are still unknown, but other than keeping the same frame, it will reportedly be a much different aircraft.
In 2015, Russian media reported that the new Blackjack would share 60% of its components with the older version.
You can find a full list of the original Tu-160's specifications here, and the Tu-160M modifications here.
Source: The National Interest
Russia unveiled the new Tu-160M2 White Swan — codenamed Blackjack by NATO — on Thursday.
Source: airforce-technology.com, The Drive