These do not mean the same thing. The two "cheers" differ - cheering can mean literally shouting at someone, and can also mean making someone feel better by giving them "cheer" (good feelings).
Cheering someone up means you are making them feel happy, usually by showing them support. (Think of it as "lifting up" someone's feelings).
Cheering someone on usually refers to sports or when you are encouraging someone (usually from the sidelines) to do a great job at whatever they are doing. (Think of this as cheering someone to "continue on" with what they are doing in an encouraging manner).
Hopefully this helps :)