Reek is about smell — a bad odour, or to give one off. Wreak is about causing damage or harm, almost always paired with a word like havoc or revenge. Memory hook: reek rhymes with leek, as in the smell of onions; wreak havoc is the fixed phrase you'll meet most often.
The core difference
Reek describes a strong unpleasant smell, while wreak means to cause or inflict something damaging.
- reek — a strong, unpleasant smell, or to give off one: His clothes reeked of cigarette smoke.
- wreak — to cause or inflict something damaging or violent, such as havoc or revenge: The storm wreaked havoc on the coastal town.
How to tell them apart
The most common slip is writing "wreck havoc" or "reek havoc" instead of the correct "wreak havoc." Wreak is a verb about causing harm or violence, and havoc is its most frequent partner, so it helps to memorise the phrase as a single unit rather than building it from its parts.
Reek, by contrast, has nothing to do with damage — it is entirely about smell, whether describing an actual odour or, figuratively, something that seems suspicious, as in "the deal reeked of corruption." If you mean a bad smell, use reek; if you mean causing harm, use wreak.