When you win a war against a vassal, they get imprisoned. When you win via white peace, you have the option to imprison the vassal, but doing so risks rebellion (no tyranny though). I usually just go for white peace against rebelling vassals since you get a significant opinion boost post crushing a rebellion and new vassals given the titles won't have that opinion boost (especially if you invite a new noble to court for the spot rather than revoke a title and give it to another vassal). The only time I fight an independence war by my vassals past white peace is when my vassals are of a different religion, culture or both. In that case, having them imprisoned post war lets me strip their primary title and redistribute power to people of my culture/religion.
If you beat a liege in a plot to revoke, the liege usually loses steps down to their successor (on muslim playthroughs anyway, not sure on christian playthroughs since I have not had this happen in a while), if you white peace, the liege just loses prestige and you get to keep the territory. Beating your liege can be hard though unless you are already about ready to depose him or you have the free cash for mercs (this is how I usually beat my liege: good economy, saved money, and mercs). Usually best to just be on your lieges good side and have your liege on your bad side (great combo to get titles, new vassals or more counties/duchies) till you actually decide to claim independence or take their title.
