One difference is that you can turn off cold shower when it becomes uncomfortable and typically you start by short cold showers. You proceed to longer ones only as you adapt. Another is that, water is more effective in cooling things then air.
If you are walking outside in t-shirt only, it is harder to get back to warm place when you suddenly feel cold or shivering. With cold shower, you just turn it off.
Well while I'm not a professional in the topic in any way, I'd personally seen the benefits of walking in the cold (e.g. in winter where people were wearing many clothes on top of each other) with just a tshirt: it makes me feel much more alive, alert (in a good way, not in an anxious way), and much more immune to pretty much anything both physically and mentally.
Funny you should say that. For a couple of years in my life, when I was about 20, I went through this phase where I went around all year long in just a t-shirt (and shoes and trousers, of course). I also took hot-and-cold showers: I showered with cold water then alternated cold and hot a few times.
I did this because I got sick and tired of catching a cold every year and having a very high fever (39 to 41 C all the time). I blamed having being overperscribed antibiotics as a child, which at the time I believed had weakened my immune system. That was probably nonsense, although it's true that when I was a kid it was common to give kids antibiotics for viral infections, so I got a lot of antibiotics for no good reason at all. I figured that exposing myself to cold would help me build up my immune system again. Also probably nonsense.
I can't tell if anything changed by what I was doing. I never felt any different from doing it, not "more alive" etc as others report. I got over my colds with high fever but that was most probably because I grew up (I think adults don't get such high temperatures from a cold as kids do). I sure got lots of strange looks and people kept asking me whether I'm a) cold and b) mad.
I didn't use any breathing techniques or meditation or anything like that. I didn't particularly exercise much, though I was a strong lad at the time (now, not so much). I think I just sort of willed myself to do it and did it and then realised it wasn't that hard after all. I think the true secret, the real lesson, is that we are made to withstand a lot more cold than we think and we're just used to being warm a lot more than we were meant to be if we lived without all our technology. Except of course we don't have to be cold because we have all that technology. So there's no reason for breathing exercises and so on to learn how to withstand the cold. That's what our brains are for.
If you are walking outside in t-shirt only, it is harder to get back to warm place when you suddenly feel cold or shivering. With cold shower, you just turn it off.