The biggest difference resides in partition function split and merge behavior. With B-Tree indexes, you can split and merge non-empty partitions. SQL Server would split or merge the data automatically, granted with the schema-modification (Sch-M) table lock held in place. Other sessions would be unable to access the table but at least split and merge would work.
This is not the case with columnstore indexes where you would get the error when you try to split or merge non-empty partitions. There are several reasons for this limitation. Without diving very deep into columnstore index internals, I could say that the only option of doing split of merge internally is rebuilding columnstore index on affected partitions. Fortunately, you can split and merge empty columnstore partitions, which allow you to workaround the limitation and also implement Sliding Window pattern and use partitioning to purge the data.
With SQL Server 2017, the logic gets a little simpler, as you can directly truncate partitions instead of shuffling them off to a separate table.