When you "remove" something, you take it away.
"Please remove your shoes before coming in."
"The detergent will remove even old stains."
"She was removed from her position as the manager."
When you "move" something, you change its position.
"I can hear someone moving in the dark room."
"We're moving to China." (relocation)
It can also mean to cause strong feelings like sadness or happiness.
"I was moved by the sad story she told me."
These are just a few examples and I hope it gives you a clearer idea.
If you need more help with your English, I'd be happy to help as a community tutor.