"Hōō" and "fushichō" are not considered the same creatures. The former refers to a Chinese-origin mythological bird, and the latter is the translation of phoenix in western mythology. Originally, "hō" is male hōō, and "ō" is female hōō.
As for "fushichō", "fushi" and "chō" mean "immortal" and "bird", respectively. Furthermore, since these meanings are provided by kanji 不死 and 鳥, the romaji (or also kana) "fushi" and "chō" have other meanings, thus, "fushi-ryū" can mean "joint dragon" other than "immortal dragon", and similarly, "chō-ryū" can mean "super dragon" or "long dragon" other than "bird dragon".