Both offer email alerts but Firefox Monitor said my old email was found in 11 data breaches while haveibeenpwned said 14.
More services like this can't be a bad thing. If haveibeenpwned disappeared tomorrow, we'd have no replacement
"Breach data provided by
Have I Been Pwned"
but it is a good service, that it alert me when i want to navigate a site, where data breach happened.
When you submit the search, there is a message below the Show All button which states that the breach data is provided by Have I Been Pwned. So it shouldn't differ at all really. Hopefully this is just the first steps by Mozilla towards creating this kind of service.