This may not be a complete list, but it's what I can think of now.
Reasons to slingshot:
1. The slide comes back to the rear slightly farther than when the slide stop/release is used, providing slightly more spring force to drive the slide forward. I've actually encountered a situation where a gun was so dirty that it wouldn't feed the first round from a mag using the slide release but would work every time when the slingshot method was used.
2. It works on just about all guns. There are some guns out there (Walther PP/PPK/PPKS, CZ52, etc.) that will lock open on an empty mag but don't have an external slide stop/release. Slingshotting is a universal technique that works on just about everything.
3. It's a gross motor skill while operating a slide stop/slide release is a fine motor skill. Don't know how much to make of this, but some feel like it's important.
4. On SOME guns (absolutely not all guns, or even most guns), repeated use of the slide stop/release to drop the slide can cause wear to the slide and possibly eventual failure.
5. On SOME guns (absolutely not all guns, or even most guns), it can be difficult to drop the slide using a slide stop/release. I've run into a few guns where the force required to operate the slide stop/release is very high and others where the slide stop/release is very low profile and difficult to get purchase on. In either of those cases, using the slide stop/release probably wouldn't be a great idea.
Reasons to drop the slide using the slide stop/release:
1. It's usually faster.
2. It is simple and can be more foolproof. There is some technique required to slingshot a slide and doing it wrong can cause a malfunction, but pushing on the slide release is dead simple. At least one manufacturer discourages using the slingshot technique and actually recommends the slide stop/release technique because they feel it's a more sure way to get the gun back into battery.
3. It is a one-handed operation. An injury or disability could make it impossible to slingshot a slide and while there are methods for one-handed racking, they are usually not quite as reliable as either normal slingshotting or using the slide stop/release.
Additional thoughts:
It's a mistake to look at the lists and say something like: "OK, 3 for slide stop/release, 5 for slingshot--an obvious win for slingshot!" The key is which ones apply in your situation. For example, if your only self-defense gun is a Beretta 92, then 2, 4 and 5 on the 'slingshot' list don't apply to you.
Not all of the listed reasons have the same level of importance. Reason one on the slingshot list, for example, is not likely to be a factor except in unusual/extreme circumstances. Some people don't give any credence at all to reason three on the slingshot list since reloading involves pushing a magazine release which is also a fine motor skill and yet people routinely manage it with no problem.
The point is that, in spite of the fact that the general problem is pretty well defined and understood, different people are likely to make different decisions due to differing circumstances. A guy who has several self-defense guns, one of which is a Walther PPK, might be predisposed to slingshot because it works for everything he uses. Another person could look at all the same factors and come to a different solution because he has a disability of his weak hand that makes slingshotting difficult.