There is not difference, it's just that "Merecer" is a reflexive verb.
"Yo me merezco un descanso."
Translates directly to the English:
"I deserve myself a break."
You can also say it this way, but it sounds a little more formal and it's uncommon:
"Yo merezco un descanso" (notice the lack of 'me')
Both mean the exact same thing though.