Just to point out, Wizards are wagging fingers, speaking some unknown words, and pulling out strange materials. Not exactly certain why that should be considered less strange than singing or poetics or what have you. And Clerics are appealing to their deity to grant them the ability to cast spells which kind of doing the same thing non-Cleric casters do. Which begs the argument as to why they should be considered deities at all rather than really powerful spell casters who call themselves deities.
They're all equally silly or not silly depending on how you choose to view it. A Bard can be viewed as being a silly, prancing idiot or a serious, words of true power caster that is activated by song. A Wizard can be viewed a silly spouter of gibberish and spasmodic gestures or a serious, words of true power caster that is activated by ancient dead language and barely remembered gestures from a long forgotten age.
For whatever reasons, people often default to either is fine for Bard and only the 2nd option is ok for Wizards. But you don't have to do that for your campaign. You want serious - choose for the Bard to be serious. You want silly - you can let Wizards be silly too. That's a campaign choice and just make sure your players are onboard with it if you're DM.
