It's not elaborated on in the film, but the narrative hints that he has a criminal past, one which his brother disapproved of. In the Ultimate Spider-Man comics that introduced Miles,
Both Aaron and Jefferson were involved in a criminal deal with Turk Barrett that went sour, resulting in Jefferson going to prison, where he was greeted by Nick Fury, who used him to infiltrate one of the local gangs in an attempt to get to Kingpin. Afterwards, he was offered a position in SHIELD, but refused and instead became a police officer. Aaron did not reform and continued performing criminal activities, eventually becoming the Prowler by stealing a powersuit from Tinkerer to protect Miles from behind the scenes.
It is likely that they kept a similar origin story where Jefferson's antipathy towards his brother is rooted in Aaron having continued being a criminal (although Jefferson likely thinks him more a petty thug). By whatever means, Aaron acquired the Prowler suit and kept that identity until he realized Miles's identity.
Some small evidence from the graffiti scene:
Aaron: "You know, me and your dad used to this kind of thing in the day ... It's true. Then he took on the cop thing..."