-How are you feeling?

Talking about your physical health, like are you sick.

or

How are you feeling emotionally? Are you very sad?

-How do you feel?

1st: Are you sick? Are you sad?

2nd:What is your opinion about something?

eg: How do you feel about ordering pizza tonight? (Yeah, pizza sounds good.)

How do you feel about the new logo design?

(I don't really like the new logo)