On my dinghy in Croatia I would simply find that a longer leg means that I hit the stony bottom more often but I guess it depends upon what kind of vessel the outboard is attached to and where you are using it?
Depends what boat it is and if it's in a well or on the transom. If it's on the transom then a long shaft helps when the boat pitches and the prop starts breathing air, the downside being that they are heavier, more expensive, rarer if buying used, and flippin' awkward things to handle and transport (especially if extra long).
It depends on the boat. For a dinghy, I can't see any advantage to a longer shaft, for a cruiser - sail or motor, a long shaft puts the prop lower, so it's less prone to being lifted out of the water in rough weather. If the OB is mounted off centre, heel in the wrong direction could deprive you of drive, just when you really don't want that to happen.
OTOH, they are, as Angus says, heavier ... and flippin' awkward things to handle and transport. If you can leave the OB on your boat with a reasonable expectation of it still being there when you come back, manhandling it is far less of an issue, so I'd go with the longer shaft, but if you have to lug it about, you have to decide whether or not the inconvenience is worth the improved drive. Of course, if the short one doesn't reach below the transom, it won't do a lot of good, so you're stuck with a long.
Whether or not to leave on the boat is a whole other debate. You can make it hard for someone to make off with it, but not impossible - it's not unknown for villains to cut out a big lump of transom so they can make off with the motor, but I suspect that's only going to happen for an expensive motor that's being stolen to order. My gut feeling is that, for mere mortals, you only have to make it harder to steal than the one on the next boat.