This is not grammatically correct, but is most likely an allusion to the Apple "Think Different" campaign, which is perhaps one of the most famous advertising campaigns in modern history. See:
Think Different
Also it is worth saying that in English there is a tendency in some circumstances to use adjectives for adverbs. I have noticed it more frequently in the American dialect. For example, here we have "real good coffee" rather than "really good coffee". Compare for example "that is a fast car" to "that car is fast."
However, let there be no doubt that "drink different" is not correct, the verb requires an adverb not an adjective and "different" is always an adjective. However, it is a deliberate rhetorical device to make you stop and think, probably thinking of that campaign.