Don’t worry; if you’re one of the many people who says “I am well” and/or “I feel differently” (both incorrect) during normal conversation, then I won’t feel good (yes, James Brown had it right all along!) about correcting you. ; )
First of all, let’s define the verb “to be" (I am, you are, she is, etc.). Yes, it is a verb, but it is not an action verb; instead it is a verb that indicates a state of being, better known as a linking verb.
For linking verbs that indicate states of being, such as “to be” and “to feel”, we should NOT use adverbs (such as "well" and "differently") to modify them. Instead we should use adjectives, to describe ourselves (people are nouns) instead of the verb itself. That’s why we say “I am hungry” instead of “I am hungrily” and “I feel happy” instead of “I feel happily.”
Another issue with the verb “to feel” is that it has another, more literal definition: to touch with your hands. If I use an adverb next to the verb “feel,” then it suggests this more literal definition of the verb.
Thus, you should not say “I feel differently” unless you are, for example, referring to your different style of reading braille. And you should not say “I do not feel well” unless you are specifically referencing an illness (or lack thereof), in which case "well" would be used not as an adverb, but as an adjective (well vs. unwell).
I feel well = incorrect unless specifically referencing a prior illness I feel good = correct I am feeling well = incorrect unless specifically referencing an illness I am feeling good = correct
I am well = incorrect unless specifically referencing an illness I am good = correct I am doing well = correct
I feel differently = incorrect I feel different = correct
Other examples of linking verbs (also known as intransitive linking verbs) that should be modified by adjectives instead of adverbs:
to seem to appear to grow (old, wise, weary, etc.) to look to remain to sound to stay to taste