There are few.
From the perspective of consistency guarantees, object storage gives fewer of such guarantees (this is seen as allowing implementations to be faster than typical file-systems). For example, since there isn't a concept of directories in object store, the implementation doesn't need to deal with the problems that arise while copying or moving directories with files open in those directories.
There are some non-storage functions that are performed only by filesystems, but not object storage. For example, suid bits.
It's also much more common to use object stores for larger chunks of data s.a. whole disk snapshots, VM images etc. While filesystems aim for the middle-size (small being RDBMs) s.a. text files you'd open in a text editor. Subsequently, they are optimized for these objectives. Filesystems care a lot about what happens when random small incremental and possibly overlapping updates happen to the same file, while object stores care about performance of sequential reads and writes the most.
This excludes the notion of "distributed" as both can be distributed (and in different ways). I suppose you meant to ask about the difference between "distributed object storage" and "distributed filesystem".