One of the prohibitions that apply on Tisha B’Ab is "Sicha," which means rubbing oil on one’s skin. The modern-day application of this Halacha is the use of skin creams, which is meant to make one’s skin feel smoother and more comfortable. The use of skin creams is forbidden on Tisha B’Ab, as it violates the prohibition of "Sicha."
However, Hacham Ben Sion Abba Shaul (Israel, 1923-1998) ruled that one may use cream to treat infected, chapped or scraped skin. In such a case, the cream is applied not for pleasure, but rather to cure damaged skin, and treatment of this kind does not fall under the prohibition of "Sicha."
It is forbidden to apply perfume on Tisha B’Ab, and it is also forbidden to smell perfume. In light of this Halacha, Hacham Ovadia Yosef was asked about the permissibility of using deodorant on Tisha B’Ab, which might appear to fall under the same category as perfume. However, Hacham Ovadia ruled that one may apply deodorant on Tisha B’Ab, for two reasons. Firstly, deodorant is used not for pleasure, but rather to eliminate odor, which is entirely permissible on Tisha B’Ab. Secondly, we must take into account the consideration of "Derech Eretz," basic dignity and refined conduct. It would certainly be unbecoming to walk around on Tisha B’Ab without applying deodorant, and this is another reason why it is permissible. This applies to both spray and stick deodorants.
(Incidentally, it should be noted that stick deodorants may not be used on Shabbat, as by using a stick one changes the substance from one form to another. Thus, although spray deodorants are certainly permissible on Shabbat, one should not use a stick deodorant on Shabbat.)
Summary: It is forbidden to apply skin cream on Tisha B’Ab, except for the purpose of healing chapped, infected or scraped skin. Deodorant may be used on Tisha B’Ab, but perfume is forbidden.