Typically, organisations like the Port Mafia are pretty good at keeping their illegal dealings at arms length so that it's hard for law enforcement to catch them doing anything. It may be all but public knowledge that they're doing all those things, but they try to leave as little evidence as possible (and have a few lawyers and other legal types in their pockets to help deal with the few bits that get away from them).
As such, often the law enforcement will look for any excuse to prosecute them, and use that as a proxy for all the stuff they can't actually be caught doing. For example, notorious gangster Al Capone was arrested for tax evasion.
So prior to getting the licence, presumably there were police detectives or the like keeping an eye on known Port Mafia members hoping that they'd see, I don't know, Akutagawa using Rashoumon to destroy a stray piece of garbage, then take him in on a charge of "Unlicenced use of ability on public property". With the licence, though, the actions of the Port Mafia are halfway legitimised by the government, and it becomes that much harder to take action against them.
