OK, here's my 2cents. What sucked me into the betta hobby was my coworker got one, which she kept in one of those hourglass shaped vases that MIGHT hold half a gallon of water. He was a lively little guy and I fell in love. That was about three years ago. I researched bettas and then bought two, which I housed in 2.5 gallon filtered tanks. I eventually got three more, which I kept in various bowls, all at least a gallon.
I kind of felt sorry for my coworker's betta, stuck in a bowl that she maybe cleans every other week if it looks nasty. My tanks were cleaned weekly, with partial water changes in the tanks and complete changes in the bowls.
One of my bettas in the tank did awful from the start. He saw his own reflection in the side of the tank, and started tearing his own tail. So I kept his tank dark, but he still did it now and again. I covered the sides of his tank to make it less reflective. Still a problem. I noticed the current from the filter was making him nervous, so I turned it way down. Still a problem. He eventually succumbed to infection. The other tank dweller is still going strong.
Of the three in bowls, one died of a tumor, and the other just passed from dropsy (which did NOT have to do with the water parameters, which were checked weekly and were absolutely fine - I've been told by breeders that dropsy is just something that happens sometimes to bettas). The third is happily ensconced in a gallon jar at work.
As for my coworker's betta, he is still going strong. He's now in a 3/4 gallon drum bowl, which she still cleans maybe every other week.
Anyway, all that to say I really don't think it makes that much difference. Just keep in mind that water changes are essential, no matter where you keep them.
And the point made much earlier in the thread about temp is a good one. It's VERY hard to keep bowls at a stable temp.
L