“Habits” and “vêtements” have the exact same meaning to me. I’d say that “vêtements” sounds maybe a bit more formal than “habits” but both are used fairly commonly.

“Tenue” on the other hand has a slightly different meaning : it is used to refer to an ensemble. A “tenue” is the sum of the “vêtements” that you are wearing (trousers and a shirt is a “tenue” or a skirt, tights and a blouse…). It is mostly used to refer to the set of clothes for a specific event.

“J’ai trouvé une tenue pour la soirée.”

“Tenue” is a set of clothes with a specific purpose: tenue de soirée, tenue de combat, tenue de travail, tenue de ski/tennis…

“habits” is mostly used for the outer layer: jacket, trousers, skirt, sweater… while “vêtements” encompasses everything: shirt, underwear, socks.