There isn't a difference.

They can be used interchangeably without much noticeable difference in my opinion! "You can use the present simple or continuous to say how somebody looks or feels now" ^^^^From an online search However, if I had to stop and give particular attention to the phrases in the way I use them.... When I see someone I hardly know I will always say "How are you feeling today" When I see someone I know I will say either choice "How are you feeling today" or "How do you feel today" I can't explain why but when I use [How do you feel today] it's a phrase I say among people with familiarity.

Hi Linlin The difference is the some than the difference between present simple (general statement at all time/ always), and present continous (right now example for people who are sick we can ask them How are you feeling today "compared with yesterday" because there is evolution in the state)

I think it is the same. But maybe "how are you feeling today" is a bit more polite.

In my opinion "how are you feeling today" means something like how do you physically feel, like sick. While "how do you feel today" could mean 'how do you feel about today?'.