I think it is helpful to differentiate between mindfulness and concentration.*

I would associate flow more with concentration. And if there is no mindfulness together with the concentration, time will just "fly by". At least for me.

*As it is done in Buddhism, where both are separate spokes of the dharma wheel.

It's an interesting thing to consider.

They seem both similar but very different at the same time.

"Flow" to me feels like allowing the mind off its leash, but having it be completely focused on one task rather than its normal state of unfocused chaos.

Whereas "mindfulness" feels more like allowing the mind to rest and become still.

In the former time slips by so quickly, and the latter time can seem to stand still, but with both time becomes meaningless.

Maybe what both have in common is this disconnection from time.

Flow can deplete you because of the singular focus. Mindfulness refreshes you because you simply are - there’s no motive for it.