Grammar is a tool to deliver the intended meaning implied by the speaker to the listener precisely (Givon, 1993). Therefore, if "think different" can convey the message that Apple wants to deliver to the public, it is grammatically correct. In other word, this answer will evaluate its effectiveness in terms of descriptive grammar.
Many argued that the slogan should be corrected as "think differently". "Think" here is a perceptually accessible verb. It is a less prototypical verb as “thinking” is a mental action, involving no discernible action. “Differently” is a manner adverb, which elaborates the way of thinking. Therefore, “think differently” requests people to have different opinions from others. However, it is not what Apple trying to convey.
As Steve Jobs clarified, the intention of the slogan was to guide what people should think, instead of how to think. "Different" here is used as an abstract noun. It asks the consumers to think about the differences on the existing electronic devices, but not their sameness. Other similar expressions also exist, such as "think big".
Therefore, "think different" is grammatically correct.