That's a distinction without a difference. The original poster wanted to know what would it be like if an independent Belarus existed, whether it's called White Russia or Byelorussia really doesn't change anything. Besides, in 1918 when Belarus declared independent, it was called the Belarusian People's Republic. So Belarus is close enough.
Assuming this is the POD, and that Belarus somehow retained its independence (very difficult to do). Let's say Poland was more successful in its war against the Soviet Union, and that Pilsudski made some kind of deal to recognize and preserve Belarusian autonomy in return for a close alliance and possible customs union. Probably Belarus is not going to be very large, probably more like the size of the Baltic states than the size Belarus is now. Here would be the major consequences as I see it.
1) The Soviets are going to hold a grudge against Belarus just as they did Poland. They may still have a Belarus SSR of their own in order to show how much better the Soviet Belarus is than the other Belarus. Like the other Soviet attempts to do this, it will fail.
2) Poland's strategic position will be improved as they have a better buffer from the USSR. Not a great buffer, but anything is an improvement.
3) Since Poland is going down this path, they might handle relations with the Lithuanians better as well, and might possibly have a rump "Ukraine" around Lvov. This would entail a decisive rejection of the Polish nationalists in favor of Pilsudski's vision of Międzymorze (a confederation/alliance of independent states). If so, it would make Poland an extremely important figure in Eastern Europe as the head of a major alliance in eastern Europe. How successful such an alliance, or how long it would last is very debatable.
4) This will change postwar European politics. France will have a strong partner in the East to "contain" Germany. It will also help keep the Soviet Union isolated.
5) If the Nazis take power later, of if any revanchist German government takes power, they are likely to treat such an alliance with far more caution that Hitler displayed against Czechoslovakia or Poland IOTL as there is a greater chance of a big war (Hitler did not expect a major war when he attacked Poland). This may delay a major European war by many years.
6) Belarus itself is likely to be similar of the other eastern European countries at the time - a dysfunctional parliament interacting with someone who is a de facto dictator. The government will move to emphasize Belarussian culture and create an identity separate from Russian.
After that, it's hard to see what happens because World War II changed so many things about Eastern Europe, and it's hard to see how WWII might or could happen with such a change.