In 'Public' edition your dumps are uploaded to public cloud. For certain enterprise, it's concern to upload production artifacts to public cloud. In 'Enterprise' edition, applications are privately installed and runs on your datacenter/local machine. Thus production artifacts will remain within your corporate firewall.
'Enterprise' edition has more features such as: capability to download the reports in CSV format, expert advice on JVM settings, automatic problem detection and fine grained solution to the problems
As shown in the above table, in the 'Public' edition there are limits on the number of dumps that you can upload, size of the file that you can upload. Where as in 'Enterprise' edition there are no such limits. You have unlimited access.
In 'Public' edition, service requests are attended within 3 days. Only e-mail support is provided. In 'Enterprise' edition service requests are attended with in 24 hours. Both Email and Phone support are provided.