The Gur Arye explains that Y'hoshua's falling into the spies' plot would reflect poorly on Moshe, whose protege he was. This reasoning doesn't apply to Kalev.
The Avodas Yisrael explains (not in answer to this question) that Y'hoshua did not want the honor of leading the people, and wanted Moshe to retain that position. (See Rashi to B'haalos'cha 11:28.) Moshe was therefore worried Y'hoshua would do something to prevent the Jews' entering K'naan, whither Y'hoshua would be taking them. [This reasoning doesn't apply to Kalev.]
Rabbi Matis Blum, in his weekly "Torah Lodaas" sheet (5773), writes that Rav Yaakov Kamenetsky is said to have answered that Kalev was married to a righteous woman, and that would prevent his falling into the spies' plot; the same, as far as we know, was not true of Y'hoshua.
Rabbi Matis Blum, in his weekly "Torah Lodaas" sheet (5773), writes that Rav Yosef Chaver answered that Y'hoshua, who descended from Yosef who had tattled on his brothers to Yaakov, might be more inclined to speak ill of the land, so Moshe saw fit to pray specially for him.
The book Hege Yona (Jerusalem 5756), by my grandfather-in-law Rabbi Yona Munk, explains that Y'hoshua, described as tolerating everyone's personality (Rashi to Pin'chas 27:16), was of a nature to be influenced by them, whereas Kalev, described as having "a different spirit" (Sh'lach 14:24), was not.