‘Carry on’ has several meanings. One of the meanings is ‘to continue one’s work or duty.’
Please do not get up. Carry on your work.
Despite all the difficulties, he is carrying on.
I am not well. Still I have to carry on the work as there is nobody else here.

‘Carry out’ means ‘to perform, to put into pratice, to give effect to’.
Whether I like it or not, I have to carry out his instructions.
You can rely on him to carry out your instructions.
He makes promises but does not carry them out.
It is easy to make resolutions but it is difficult to carry them out.