The correct sentence is
I do think very differently
because differently is an adverb, and it describes the verb 'do'.
We use the question word (how) to ask about the adverb of manner.
How do you think? very differently.
But different is an adjective, and it it describes a noun.
(1) Each person has his/her own opinions. So we say: People do not all have the same ideas. People think differently about things. As the first poster pointed out, we want to say "how" people think, so we need to use an adverb to modify the verb.
(2)_ But here in the United States, sometimes people in conversation say, "Think different."
(a) For example, let's say that our company is not doing well. We do the usual things to get customers: advertise on television, radio, and in newspapers; send fliers to people's homes; use telemarketing, etc. But business stays bad. So one day the boss says: Ladies and Gentlemen, I want you all to think different.
(i) The boss is not interested in "how" his workers think. He is interested in "what" they think. In other words, he is saying: Think hard and give me some different ideas that will help us get business.
(b) In the United States, "think" + adjective is very popular in the advertising world.