I started using them mostly for the cold weather (Janjis). It's been a revelation. Previously, I'd wear "normal" pants and would always feel somewhat cold at around 0C. Janjis have been perfect for that and lower. They're light compression, for warmth.
I also like how they feel. They're less distracting than normal pants.
As for recovery, I do wear them (or other more recovery-oriented leggings) after hard runs. I think they help; I haven't felt sore on my legs in a while. But it's hard to say if it's just because of the pants; I might just be used to it.
Finally, I haven't felt a difference in performance with performance oriented leggings (2XU). I know they're supposed to work, but I haven't proven that yet. Haven't ran a long (half+) race with them yet to be fair.
When I started training longer than 10K, I was seeing signs of shin splints- tender shin bones and tight calves. I started running with calf sleeves and it made a lot of difference both in reducing the pain and helping me recover faster. I also started doing calf and ankle stretches (downward dog is the best for this!) and just in general I've adapted better to running over the last few years, so now I don't use the calf sleeves even though I'm running marathon distances.
I wear full length ones when the outside temps are below 10C (~50 F). No discernible help with muscle soreness or fatigue.
I found this study (here which says there might be some benefit wearing them after a run to reduce recovery, but no benefit during. I have a pair so might give it a go on my next long run.
I generally do not run with them however when I was training for and running longer distances I would tend to wear them on those longer runs (15+ miles) mental or not , not sure but I felt like they help with fatigue.
Also in colder weather I've worn them to just keep my calves warmer / seemingly help with early tightness.
For me personally they're not like night and day difference but they do have use case that seem to help.
I've run ultras in the past and i honestly found wearing then down around my ankles and only pulling them up for brief recovery walks or after any race felt way better to me.
To clarify, i was talking about calf sleeves. I've never noticed a diff between shorts vs leggings.
Mine were as well. I used to play soccer so i didn't really have the typical runners calves so they were tight as hell on me which is probably why i never enjoyed running in them for longer distances.
This is a question I'd be interested in hearing the answer too as well. Been having issues with the muscles that run up to the calves recently, been looking at tights or the lower leg things only but no idea if they actually do anything.