"I have many / a lot of / lots of things to do today"
In this sentence, they all mean the same thing. Perhaps "lots of" sounds a bit more informal.
There are some expressions in which it sounds more natural to choose one of them
e.g. "I have lots on today" = I have a busy day today (colloquial)
"Many people suffer from diabetes"
Again you can replace "many" with the other options, but in writing, many sounds more formal and appropriate.