I’m a developer, but not really an user of Zotero - so my knowledge upon Zotero is only limited. 
I think JabRef is stronger in it’s grouping, searching and (table) sorting features.
It is possible to customize the set of “required”, “optional” etc. fields for standard entry types and it is also possible to create new non-standard types.
It is directly possible to fetch the bibliographic data (and the fulltext PDFs) by just providing an identifier such as a DOI, ISBN, … - as far as I know you’ll always have to lookup the source in the browser first in Zotero.
And - for me - the most important aspect is that JabRef is using a simple storage format (bibtex) which can be stored anywhere in the file system, can directly be used in LaTeX (without an additional export step), can be shared directly, can be simply versioned in version control systems, and can be even used in conjunction with SQL-Databases.
So to come back to your first post:
Are JabRef and Zotero trying to solve different problems?
I would say no, but both solve the same problem differently
You can also find an objective comparison of Zotero, JabRef and various other tools here provided by the Library of the Technical University Munich.