Why does the Torah switch the word for men from כְלִי to שִׂמְלַת for women?
The Malbi"m there brings down a dispute over the meanings of the two terms. Either כלי and שמלה are synonyms and both refer to articles of clothing, or כלי is a broader term that includes clothing (such as שמלות) and other things. He brings extensive proofs for the opinion that differentiates between the two.*
The phrases in which the two words are found are actually quite different. One is
ילבש שמלת אשה ~ shall wear a woman's garment
and the other is
יהיה כלי גבר ~ shall be a man's instrument .
The verb in the latter phrase is semantically much broader in that it refers merely to "being upon" a person, rather than specifically wearing, which indicates that the object of the verb, "instrument," as opposed to "garment," could also be more general. Additionally the term for "garment" in this case is even more specific than the more common term for garment (applied to this exact set of prohibitions by חז"ל), i.e. בגד. Instead the Torah uses שמלה, which was used recently in the phrase שמלת שביה, contextually associated with an object of beautification. Finally כלי גבר is lexically associated with כלי גבור מלחמה ~ the instruments of a warrior.
All of this is to argue that a man is prohibited from wearing (specifically provocative) clothing of a woman, whereas a woman is prohibited from putting on (specifically war-like) items that would normally be used only by men.
*I would point out that לפע"ד, the second opinion could come לבאר ולא לחלק, but there is no harm in separating them.
The sefer Ohel Moshe here brings the following explanation of R. Nosson Adler:
The reason why the beginning of the posuk does not say "a woman shall not wear a garment of a man" using the same structure as the second half of the statement, is because it is well known from our holy seforim that one who constantly thinks about sexual immorality will be reincarnated as a woman.
Therefore the posuk says לא יהיה כלי גבר על אשה, because the word כלי can mean thoughts - from the word נכליהם (Bamidbar 25:18), and thus the posuk means “the thoughts of a man should not be about a woman”, and thereby “a man shall not put on the garment of a woman” - he won’t be reincarnated as a woman.
Translation taken from here.