What's the difference between a cosponsor and a multi-sponsor for a bill?

Co-sponsor means just two sponsors. Two members of the legislature may have originally introduced a bill.

Then, after that, others may "jump on board" and be sponsors too. That will make it "multi-sponsored".

When you have a bi-partisan bill, one that both political parties support, having one member of each political party sponsor a bill (co-sponsor) often increases it's chances of being reviewed, and voted on.

Most bills written up by legislators, luckily, never see the light of day. They get pigeon-holed, and disappear. The more legislators who will sponsor a bill, the better chance that a committee chairperson, will give that bill a hearing. Without a hearing, and vote of entire committee, new bills don't get voted on by whole legislature.

It's part of checks and balances that help make the system work better, so most legislators won't get bogged down reading, voting on insignificant, or poorly written bills.

A co sponsered bill is one with only two sponcers and a multi sopncered bill is one with many sponcers. Other than that there is no difference.