Climactic is the adjective corresponding to climax. Climatic is the adjective corresponding to climate. The antonym of climactic is anticlimactic (one word, no hyphen). Anticlimatic could theoretically work as a word (to describe someone or something that opposes climate), but it’s usually just a misspelling of anticlimactic.
Examples
The misuse of climactic in place of climatic and vice versa is common—for example:
[H]e was there to get noticed and add some spice to what would have been an otherwise anti-climatic political exit. [Los Angeles Times]
Corley gave a touching performance in Falstaff’s climatic scene when the King banished him just after coronation. [Pegasus News]
According to data from the NOAA National Climactic Data Center, there has been above-average fire activity in parts of Oklahoma, Texas and New Mexico. [Huffington Post]
And here are a few examples of the words used well:
Other than any potential climatic impact, the effects of reduced sunspots would have little direct impact on humans. [Sydney Morning Herald]
It’s only when we reach the final, climactic standoff that First Class begins to show its true potential. [CBC]
The second act was one long, anticlimactic letdown. [Guardian]