a major reason was deterrence. if opposing factions know you have a WMD, they're far less likely to revolt. of course, they had to demonstrate its powers at least once, otherwise it would be dismissed as a hoax or propaganda. as for Alderaan, that place was thinly populated, having a global population only in the millions, and was an absolute hotbed of Rebel support. it was unfortunate that they had to waste such good real estate in the process, but I can see their reasoning. of course, many sentients joined the Rebel cause in direct response, but that was a miscalculation on their part.
keep in mind also, as we saw during the Battle of Endor, the Death Star didn't have to use all-out destructive power. it's capable of firing much more frequent underpowered shots, which are ideal for taking out capital ships.
in general, the Empire was top-heavy, and followed the philosophy of bigger = better. if nothing else, building a space-station/superweapon was in character for them.