Generally speaking, when watering a vegetable garden, are there any benefits to using cold water vs warm or vice versa?
I'm in Wisconsin, so 90+°F is about as hot as it gets around here. Typically the temps are in the 80s during summer months. I'm mainly comparing water that has been sitting in a brown rain barrel in the sun (warm) vs water from a hose (generally cold).
It depends -- but it probably doesn't matter.
Steve Solomon discusses correlations between watering and temperature in "Gardening When It Counts" (p123), though he doesn't exactly cover your question. Watering lowers the soil temperature because evaporation is a cooling process. I'm not sure whether hot or cold water will cause more temperature loss (seems like that's a question for Physics.SE). I suspect (but the physics people would know better) that there's not much real difference at the end of the day if you're watering the soil and not the foliage.
When you think about it, you're going to put down say 1" of water and expect it to soak into the top 12" or so of soil, then you're mixing each unit volume of hot water with 12x the volume of soil. The temperature delta between the soil and the water isn't that big, so you're really not going to modify the soil temperature in a significant way. (And then you'll lose some temperature to evaporation anyway.)
If you're wetting the foliage, water above a certain temperature is going to cause leaf damage. I don't know what that point is, and I suspect it would be different for different plants. So I'd recommend not using the hot water for foliar feeding.
I have been growing a wide range of vegetables for many years, and have never deliberately used warm water (although the water in the hose is sometimes warm after lying in the sun for a time); I very much doubt that there are any benefits at all in using warm rather than cold water, or vice versa, except with tender seedlings, before they are hardened off, when I always use tepid water. Soil and air temperature, of course, are critical to growth rate and success.
First off, without getting into the real science (most of which is beyond my knowledge), it's "generally" considered bad practice (for the health of the water within) to keep a rain water barrel in full sun.
Coming to the question, your temperature is definitely hot enough to considerably warm up the water inside a rain barrel exposed to full (summer) sun. Personally I wouldn't use warm water (if cooler water was an available option) on vegetable, shrubs or other plants in the garden. But if I had no other choice, then of course I would use it (via watering at soil level, not directly on the foliage).
I was always told to use cold water when watering plants etc. I never really questioned "Why?". I have read about using (and have used) "tepid" water (as per Mancuniensis' answer to this question). During the heat of the summer here in Missouri (think ridiculously hot) I always discharge the hot water that has built up in my outdoor hose before watering the garden (with cool water).
If you cannot move your rain barrel into the shade, I would suggest something like filling watering cans, buckets, etc. from the rain barrel, then putting them in a shady area for a day or two before using that (cooler) water in the garden.
You should use cooler water whenever possible. Water holds greater amounts of carbon dioxide at cooler temperatures and since roots love carbon dioxide it becomes beneficial for the whole plant to use COOLER water. Do not use very cold water as it will shock your roots/plant.
