They're often called jet streams, but the visible bands you see in Jupiter are probably more like Hadley cells. Convection currents that move heat from lower in the planet's surface to higher before cooling and circling back downward. The darker bands are rising warmer atmosphere, the lighter bands, made lighter in color by more ice, are falling.
Jupiter's fast rotation gives it several Hadley cells per hemisphere. By comparison, Earth has 3 and Venus has 1. Faster rotation and perhaps Jupiter's larger size as well creates more Hadley cells.
That said, if Jupiter didn't rotate, it would still have heat from below, rising and escaping. It would still shed temperature from below, because of internal heat.
What it might look like is a complicated question but I think a uniform brown is unlikely. It would still have regions of darker, warmer rising air and lighter more ice filled, colder falling air. Jupiter's bands aren't because it's atmosphere doesn't mix, it's because warm Jupiter air is a different color than cold more icy Jupiter air.
Also, no rotation effectively gives the planet a sun side and a night side. That would vary some depending on no rotation at all, vs a slow tidally locked rotation, but even so, the Sun side would be warmer. Jupiter's upper atmosphere is more driven by heat from below than heat from the Sun, but the Sun would still have an effect, especially if it only shone on one side of the planet.
I don't think we have a good sense of what non-rotating Jupiter type planet might look like. The upper atmosphere temperature would be a primary factor, but with no rotation, I think it would probably require some high level mathematical modeling to work out an answer.
Some kind of grand current might be possible, perhaps with patterns perhaps similar to the "Y" shape observable in UV in Venus' atmosphere, or perhaps an ocean current like pattern, but that's just speculation.