Everyone's assuming that this is "think differently" with the -ly dropped off, but note that there are also formations like think fruit or think pink , and "think different" could belong to that group.
In other words "What should I think about this product?"
"Think (that it is) different".
Anyway, "think differently" is a terrible slogan.
english.stackexchange.com/users/None/
Merriam-Webster claim
that different as an adverb dates at least as far back as 1744.
english.stackexchange.com/users/39/nohat
Well, it's certainly not the first time an adjective is used as an adverb in American English.
I'd call it informal, but not necessarily ungrammatical.
english.stackexchange.com/users/706/user706
One possibility we're forgetting here is the that the adjective may be a substantive adjective .
Substantive adjectives are adjectives which are used alone without the noun they are describing.
For example, good , bad , and ugly in this sentence, 'The good , the bad , and the ugly , which is really, 'The good people , the bad people , and the ugly people .'
In this case, Apple's slogan, 'Think different', would be read 'Think different things', and is thus a perfectly grammatical.
english.stackexchange.com/users/1065/J D OConal
Apple did not mean "think differently".
That is, they are not suggesting that you think in a different way.
They really meant "think different", that is, rather than thinking about the things you usually think about, think about things that are different.
An analogous slogan might be, "Thinking about your opportunities as a high school graduate?
Think college."
english.stackexchange.com/users/11928/David Schwartz
"Think differently" would mean: Please think in a way that is different from the way that other people are thinking. "
Think different" means:
Think about things that are different, or how to do things that are different.
The slogan doesn't tell you how to think, but what to think.
english.stackexchange.com/users/73713/gnasher729
If the message is taken to mean: Think "different".
Then it could be grammatically incorrect.
Just not sure I can say this is "punctuationally" correct, but from a logo/branding/marketing point of view the use of punctuation may take something away from the impact of the message even if making the intention of the message unclear.
english.stackexchange.com/users/59070/Rick Henderson
Not too long ago, Apple Computer used the phrase "Think Different" as an ad slogan. Is this a grammatical error (that is, it should be "Think Differently"), or are they trying to say something else (and what would it be)?
Is Apple's Old Slogan, "Think Different", grammatically incorrect?
english.stackexchange.com
<phrases><grammaticality><slogan>
english.stackexchange.com/questions/2632
english.stackexchange.com/users/61/Chris Dwyer
Apple were not intending to say "Think differently" but instead encouraging people to think about "different things". Indeed, it's possible that the word "different" may be used as a substantive adjective. Therefore, there slogan doesn't tell you how to think but what to think. The "different" is being used here as a substantive adjective, as in "Thinking about your opportunities as a high school graduate? Think college."
Apple were not intending to say "Think differently" but instead encouraging people to think about "different things". Indeed, it's possible that the word "different" may be used as a substantive adjective. Therefore, there slogan doesn't tell you how to think but what to think. Apple did not mean "think differently".  They really meant "think different", that is, rather than thinking about the things you usually think about, think about things that are different.  
Apple really did mean "think different". That is, rather than thinking about the things you usually think about, think about things that are different.   Apple's slogan might be informal but it is not considered to be grammatically incorrect. Furthermore, the slogan invites you to think about "different things" rather than to think "differently".
It means that you should think their products are different - and it sounds better as a slogan. Adjectives are often used as an adverb in American English in an informal way. The "different" is being used here as a substantive adjective, as in "Thinking about your opportunities as a high school graduate? Think college." The slogan doesn't tell you how to think, but what to think.  
Apple did not mean "think differently".  They really meant "think different", that is, rather than thinking about the things you usually think about, think about things that are different.  
It might be informal, but it is not necessarily grammatically incorrect. Apple's slogan invites you to think about "different things" not to think "differently".