RoundCube is the more common of the two in our index: it runs on 28 websites against 4 for Zimbra. Counts come from real scans of each site's home page, not vendor claims, and cover the sites in TechLookup's index rather than the whole web.
RoundCube is free and open-source web-based IMAP email client.
Zimbra is a provider of open-source email and collaboration software, offering solutions for email, calendar, contacts, and collaboration for both individual and enterprise use.
Within the webmail sites we have indexed, RoundCube holds 0.0% and Zimbra 0.0%.
Want to know which one a particular site uses? Look it up and see its whole stack, or compare two sites side by side.