These cases are not the same. In terms of the hierarchy of republics in the Soviet Union, Belarus was on the same level as Russia, Ukraine, Kazakhstan, Tajikistan, Armenia and so on. All these "first-level" republics became independent without exception.

Russia, however, was unique because it was a "Federative Soviet Republic" in the Union. This means that Dagestan, Tatarstan and others were actually at the "second-level" in the hierarchy. While there was always a mechanism for the immediate republics of the Soviet Union to secede, there was no such provisions for the republics of Russia even though it was still within the Soviet Union.

As has been mentioned, Belarus was one of the fifteen union republics that constituted the USSR and became independent in 1991 (or 1989 for the Baltic states).

The other republics you mention (Tatarstan, Chechnya, Dagestan, etc) are autonomous republics, in that while they have full governments, they are still part of a larger body, in this case the Russian Federation (the official name of Russia, in part because it has 85 administrative units, including 22 republics; these numbers include Crimea).

This was not a unique feature to Russia: during the Soviet era states were created for various nationalities and given varying levels of autonomy. As noted, fifteen groups were given their own union republic, the highest level. Then came autonomous republics (these changed over the decades the Soviet Union existed, so I'll note the ones that were around at the end); most were based in Russia, but Georgia (Abkhazia and Adjara) and Azerbaijan (Nakhichevan) each had one as well. Finally came autonomous oblasts, the lowest form of autonomy. There were again mainly in Russia, but also Georgia (South Ossetia), and Azerbaijan (Nagorno-Karabakh).

So in short, Belarus had been a full-level republic during the Soviet era, so like the others it became an independent state.