If I'm playing the Romans, I might occasionally bribe an allied Roman army if it consists of units that I really want -- such as triarii, before I'm able to build them myself. Otherwise I don't bother bribing them just because "they're cheaper" -- because they're really not that much cheaper, when you consider that the real cost is in the upkeep.

The real advantages are more like the advantages of mercenaries -- they might already have some experience chevrons, they might be units you can't build yet, and you don't have to wait for "build time" to bring them into play. These are the most important factors to take into consideration when expanding your army with bribes.

A more useful way to spend your bribe money is buying new family members to have for more generals, governors and (most importantly) more children! Another common use for bribing armies is when I need to eliminate a rebel or enemy army from one of my territories but the nearest army can't afford to abandon their garrison duties to attack it for a little extra experience. And the third and best reason for bribes is when I just don't feel like a rinky-dink battle with a small rebel peasant army and want to move on with the game.

Bribing an allied Roman faction's armies on the grounds that it reduces their military strength is ridiculous because it has absolutely no net gains for you. Bribing an enemy Roman faction's armies can be extremely useful though.