No notable difference.

The second is kind of old-fashioned and the only exemples that come in mind are proverbs like « Rien ne sert de courir, il faut partir à point. » − La Fontaine

More generaly, the second is rarely (never?) used without another proposition after, whereas the first could be a sentence alone.

And whatever is the proposition after you could still use the first construction, which is, again, more common nowaday.

“Ça ne rime à rien de…” synonymous with “Ça ne sert à rien de…”?

Absolutly. Exact same meaning. I don’t know the reason and I can’t find any other verbe than “rimer” that could fit here (to replace ‘sert’)…