The only clear advantage is when the text string to be indexed can be quite long. Then the hash index can be much smaller and therefore better cached. In the extreme, inserting values longer than about 2700 bytes can error-out altogether with btree indexes, but will still work with hash indexes.
With shorter strings, hash indexes have more disadvantages than advantages. In addition to not supporting ordering, they also don't support unique constraints (and thus can't be on the receiving end of foreign keys) and in the non-unique side they tend to become unbalanced if some values are very much more common than other values. Also, being less used means they are less well-tested and so more likely to have undiscovered bugs; and less work has gone into optimizing them.
Also, I think the metapage of hash indexes is still a point of contention for insertions, which can lower performance.