Short simple answers that I have always gone by:
Skirmish= Short, simple, run and gun games sometimes with brief scenarios that end either by time limit or by sure winner. Usually skirmish games are 5 minutes to 30 minutes in length.
Milsim= Longer, more drawn out scenario games, usually with multiple respawns, command structure, scenarios that lead to other scenarios that lead to a common end goal. Milsims are usually non stop the entire day long, 8 hours to 24 hours in length.
Then theres a term I use,
Skirmsim= Healthy mix of both, you run and gun the players early to warm them up, then break off teams, give them some missions to do, assign a couple of leads or communication liaisons and run 1-2 hour games, but keeping them tight and fast paced.