"Don't exert yourself much," is more focused on the effort of anything (i.e. sport training, lifting heavy stuff, etc.). "Don't overwork yourself," is more related to doing too much of a task like working at a job, helping too many people, etc. Exert is more short term whereas overwork could be over a longer time period.

The first sentence is better stated "Don't overexert yourself" or "Don't exert yourself too much" The second sentence is correct. Both have pretty much the same meaning.