What happens if the seller keeps operating a similar business right next door after promising not to?

If the purchase agreement, or a related agreement, includes a valid non-competition covenant from the seller, operating a similar business nearby in breach of it is a breach of contract, and you generally have remedies including a damages claim and, depending on the circumstances, an injunction to actually stop the ongoing competing activity rather than just being compensated for it after the fact.

It's worth knowing why a seller non-compete like this can even exist: the Employment Standards Act's general 2021 ban on employee non-competes carved out a specific exception for a seller who becomes an employee of the purchaser as part of a business sale, which is the main route sellers' non-competes remain enforceable at all. Even where that exception applies, the clause still needs to meet ordinary common-law reasonableness limits on its scope, geography, and duration to actually hold up if it's challenged. Review the specific clause and how it was structured before assuming it's automatically enforceable as written.