Basic Search Center V.S Enterprise Search Center
The differences between Basic Search Center and Enterprise Search Center really are the abilities to customize.
In Basic Search Center, you can only edit the existing search page and search result page and you cannot add more pages of you own to it.
In Enterprise Search Center, you can author you own pages (tabbed search pages). Enterprise Search Center only available in SharePoint server (standard or Enterprise), also you need to enable the following features for it:
SharePoint Server Standard Site Collection Feature
SharePoint Server Standard Publishing infrastructure Feature