Did any achronim write books comparing Arabic to Hebrew?
Several rishonim draw comparisons between Arabic an Hebrew. But earlier, even the Tannaim used foreign languages to explain difficult Hebrew words. See, for example, Bavli Rosh Hashana 26a (here, the six last lines). The early Hebrew grammarians used Arabic (their native language) in constructing the formal grammar of Hebrew, borrowing the concept of triliteral roots (and of using פעל as the base word for verb patterns) and other ideas from Arabic. Rav Saadia Gaon used Arabic to explicate difficult Hebrew words in his commentaries on Tanakh. For a treatment of this in particular, see Professor R. Steiner's new book on Rasag, "Biblical translation in the making." Also, the Arukh (by Natan b. Yechiel of Rome) frequently compares Aramaic words to Arabic and other languages. However, strictly speaking, no, I don't believe there are any books by a rishon or acharon that are solely on the similarities between Hebrew and Arabic.
I do not know of any Achronim, but some Rishonim (such as Rashi and the Ramban) discuss similarities.
Rabbeinu SaAdiyah Jaon compared the 2 languages in his commentary on Sefer YasSirah. He is a Jaon not an Acharon though.