Why Wont My FPS Go Above 60?

Bandicam should not be limiting your framerate while it is not in use. In Minecraft go to Options -> Video settings to make sure that VSync is "Off" and that Max Framerate is "Unlimited".

I doubt it will be having an effect if you're sure you've closed it, but you can also check the framerate limit in Bandicam in the FPS tab: