Not sure exactly what your question is, but both sentences are correct and common, and depending on context could mean the same thing. "things to do" is more general and include errands as well as other things, or may not include errands at all. "Errands" tend to refer to relatively short, discrete tasks outside of the home (examples: going to the store, stopping by the bank, picking up the dry cleaning, etc.). "Run" is the usual verb associated with errands, though it is not literally "running."

Errands are most associated with undertaking a short trip to collect something(s), especially on somebody else's​ behalf.