The two kinds are quite similar: Classes map to tables. Columns map to
fields. Instances map to rows. Object references map to foreign keys. In
fact, ORMs try to map the two models as closely as possible. They are quite
successful with that.
The difference is mostly in the way the database is used. The difference
is in "style" and in the patterns that are used.
Object databases are basically used like you would use an ORM on top of
a relational database. You can directly store objects (or object graphs)
into the database without the need to write an
statement. The interface is often not SQL.
It is hard to differentiate the two because they are so similar. Please
leave a comment if you are interested in certain (concrete) aspects.