Climactic relates to a climax — the climactic scene of a film. Climatic relates to climate — climatic conditions. The extra C belongs to the climax word, and the two are frequently swapped in writing about both films and weather.
The core difference
A dramatic peak and the weather.
- climactic — of a climax: the climactic moment.
- climatic — of climate: climatic change, climatic zones.
A third word
Climacteric is a further relative meaning a critical period, and in medicine the menopause. It is rare, and mostly worth knowing so it is not read as a misspelling of one of the others.
The spelling logic
Climactic comes from climax, and the X becomes CT — the same shift as in reflex/reflective. Climatic comes straight from climate, with no extra consonant to explain.