They can mean the same thing. However, the second example uses the present participle form of the verb, which tends to emphasize that the action is going on RIGHT NOW. The second sentence feels more urgent. If you had just said "I do not feel well," it could describe a usual state of being. I.e. - depending on the conversation, you might be saying that you do not feel well most of the time. However, saying "I am not feeling well." Emphasizes that you are referring to how you feel in that specific moment, even if you generally feel ill. Another example: I run (probably refers to a regular running as exercise) I am running (The speaker is running as he or she says this) And yes, it should be "feeling," not "felling."

There is no difference - they mean the same - except in the second sentence it should be 'feeling', not felling.