Scenario is sometimes used in punter site advertising such as 'You will play 6 scenarios during the day', e.g. they have 6 different zones each with a theme.
Scenario paintball (In my opinion as a scenario player and organiser) is a game that is defined by the events theme, and has missions, objectives and scoring based on that scenario theme.
If it has a theme but the game is just about going out to shoot each other and capture/hold ground then thats just paintball.
If the theme is integrated into the game, such as having to find items, putting them together to make something or gain information, holding places at certain times to score points etc then the scenario theme is core to the game.
You can find yourself in unusual circumstances in scenario, e.g. having to find a heart cushion, crawl through a corn field and call out love messages to Rapunzel (AKA a bloke in a dress) then climbing up to defend yourself and Rapunzel from flashbangs in a tower.