No, a sentence like your first example is not correct.
The second sentence correctly uses the adverb "differently" to modify the verb "think":
I always think differently from other people.
"As different" cannot replace "differently" in the sentence above, because "as different" does not act as an adverb. The sentence you wrote in a comment uses "as different" in a distinct way. Here "different" does not show how he defines himself, but what he defines himself as:
The young boy seemed to strain to define himself as different from his brother.
Here the young boy provided a definition of himself, and the definition was "different from my brother." Perhaps the boy pointed out that he and his brother have different interests and habits. If we replace "as different" with "differently", the meaning changes:
The young boy seemed to strain to define himself differently from his brother.
This sentence means that the boy was defining himself (the boy) in a way that was distinct from how his brother had defined himself (the brother). Perhaps the brother defined himself (the brother) using a quiet voice and short sentences, and next the young boy defined himself (the boy) using a loud voice and long, descriptive sentences. In this case, we do not know whether the definitions are similar or different, only that the ways the boys defined themselves were different.
Consider these two sentences:
Jin described himself sadly.
Jin described himself as sad.
The first sentence means that Jin described himself in a sad manner, such as speaking in a sombre tone or with tears in his eyes, but the sentence does not provide information about the actual content of his description. He could have been describing his height, his interests, his political positions, etc. We don't know what description he provided, but we know that he seemed sad at the time.
The second sentence means that Jin said he was a sad person, but the sentence does not tell us how he provided this description. For example, he may or may not have seemed sad at the time. It is even possible that he was in a good mood when he gave the description but considers himself sad in general. We don't know anything about his mood at the time, but we know that he choose to describe himself as a sad person.
In the sentences above, "sadly" and "as sad" perform different grammatical functions and have different meanings. "Differently" and "as different" follow the same pattern, so while both may be used in separate contexts, they are not interchangeable.