Vary is a verb meaning to change or differ, while very is an adverb used to intensify another word. If you can swap in "change" and the sentence still works, you want vary; if you can swap in "extremely," you want very.
The core difference
Vary is a verb about change; very is an adverb about degree.
- vary — to change or differ from one instance to another: Prices vary depending on the season.
- very — used before an adjective or adverb to intensify it: The soup was very hot.
- very — emphasises that something is exactly the thing referred to: This is the very book I was looking for.
How to tell them apart
The two words differ by a single letter but do entirely different jobs: vary is something a thing does (a verb), while very describes how much of a quality something has (an adverb). A quick test is to try replacing the word with "change" — if that still makes sense, it should be vary, not very.
The mix-up usually shows up in fast typing, where the "a" and "e" get swapped without anyone noticing, since both words are short and common. Reading the sentence aloud tends to catch it, because "her mood can very wildly" simply doesn't sound right, while "her mood can vary wildly" does.