What do c-1 and c-0 tags mean?

Tasks marked c-0 were supposed to be solved in order to get the Stargazer Certificate.

Now they are just left as is, and this certificate is unobtainable for new users.

For solving 120 c-1 tasks yourself you can request the Certificate of Zealous Coder.

So, these tags are only related to this site.

Do these differ in difficulty?

It doesn't matter (as stargazer certificate - for which c-0 labels are retained - is not issued to new users).

However practically yes - c-1 marked problems are somewhat more demanding.

Actually the need for newer type of certificate was due to too many simple problems collected.

Details are on this page.