|Hold it right there, outlaw!|
This article is littered with spoilers, so I reckon you might ought'a mosey on down the road if you don't want to read any plot details.
Unlike most other outfits in the game, the Cowboy Outfit does not give any bonuses or benefits to the player. It is the outfit Marston starts with in the game. It is replaced by the Rancher Outfit as John's default outfit during the mission The Outlaw's Return.
John Marston wears a long-sleeved beige shirt with the sleeves rolled up, a black denim jacket with the sleeves cut off to the shoulders, black gloves, a satchel that lays on his right side, and pin-striped pants that are worn over the boots.
When playing as Jack, the outfit changes to a beige jacket worn over a plain white shirt with a red neckerchief around Jack's neck, and tighter, darker pants tucked into the boots.
The style of hat Marston wears is a Light Center Crease, Modified-Cattleman (From a question answered by the Rockstar Support).
Overtime Jack's version seems to gather more dirt and mud on his pants then John's, but doesn't appear to gather visible dust on the jacket like John's does.