Separated by a single vowel, and in most English accents that vowel is barely pronounced in ordinary speech. Which is exactly why the mistake survives proofreading. You hear yourself say the sentence correctly and your eye slides straight over the wrong letter. Unlike some usage questions, there is no grey area here at all — one is about time, the other about comparison, and they never overlap.
An adverb of time or sequence. It answers "when" or "what next."
A conjunction used in comparisons. It only ever appears when two things are being measured against each other.
Then and than were once the same word, and they only settled into separate spellings a few hundred years ago. In quick speech the vowel is often the same, so people write the one they hear.
The rule is short. Than is for comparing: taller than, more than, rather than, other than. Then is for time and sequence: first this, then that; back then; if it rains, then we stay in.
A useful check is to ask whether two things are being measured against each other. If they are, it is than. If the sentence is telling you when, or what comes next, it is then.
This draft is much better then the last one.
✓This draft is much better than the last one.
Two drafts are being compared, so than.
I would rather walk then wait for the bus.
✓I would rather walk than wait for the bus.
"Rather than" is a comparison between two choices.
Finish the report, than send it to the client.
✓Finish the report, then send it to the client.
This is a sequence, one step after another, so then.
Quick way to remember
Ask whether the sentence is comparing two things. If it is, you need "than" — comparison and "than" both carry an "a." If the sentence is about time or what happens next, it is "then," which shares its vowel with "when."
Is it "other than" or "other then"?
Other than. It compares one thing with everything else: "nobody other than Sam knew."
What about "if ... then"?
Then. In "if you are late, then call ahead", it marks what follows from the condition, which is a sequence, not a comparison.