InVinoVeritas
Adventurer
Well, you'd have to consider the effect of spells like "Speak with Dead" and other such divinations on the investigation aspect of the campaign. I expect you'd certainly see magic being an element in the crime the players investigate as well. As you're PCs would be working under official sanction of the city they'd have both benefits and drawbacks. For one, they'd couldn't just loot the perpetrators they'd catch, but on the other hand they'd probably have equipment provided to them by they're superiors.
That's how I did it in my police game. The PCs couldn't loot, but they could requisition equipment. As a side effect, though, they could end up getting enough rope to hang themselves (read about the effects of all the firebombs one PC asked for).
Another big change was the social effect. In a typical adventure, the PCs are strangers to the community, and don't get any special treatment. As police, though, they had immense social power. They could arrest people, search places, trump up charges, frame townsfolk, and essentially, loot anyone (despite what I said above). A couple of times, the PCs had the chance to abuse their power. Since this was Ravenloft, in my game, how you treated people was important; corruption could have lasting effects on your soul. In a more typical campaign, that limit might not be there, and you and your players will have to decide what sort of cops they will be.