Difference between TAKE UP and TAKE ON (= to begin to do something)

Well. Take up means:

1. to start being keen on smth, to begin doing smth on regular basis.

• I took up tennis.
• I wish I took up mathematics.

And second relevant definition is

2. to consume or occupy.

• The books on finance take up three shelves.
• All my time is taken up with looking after kids.

Whereas take on means

1. to begin to have or exhibit.

• In the dark, the teddy bear took on the appearance of a fearsome monster.

Another definition is
2. to assume responsibility for...

• I'll take on this project if no one else wants to take it.

And I guess, another appropriate meaning of take on is to attempt to fight or compete with.

• I don't recommend taking on that bully, since he is bigger than you.

Therewith, my answer to your question is

You look exhausted. I think you've taken on more than you can handle.

Because it kinda means to assume responsibility for. I guess you see what I mean.