I wondered if there is a difference between the words שֵׁבֶט (rod, stick, staff) and מַטֶּה (rod, stick, staff) in matter of function, symbolism, and 'spiritual meaning'? A mateh seems almost like a magical wand, for Moshe used it to perform miracles and fight off Amalek, so I figured out that there must be some symbolism or deeper meaning to it.
The Radak in Sefer HaShorashim (see p. 245) on the root שבט seems to use the two words interchangeably but he does write -
"ונקרא הראש והמושל שבט כי דרך המושל לחיות שבט בידו ועוד כי בידו שבט מוסר העם שבט מושלים"
"And he is called the head and the leader of the tribe because it is the way of a ruler to live with a stick ('shevet') in his hand, and furthermore, because a stick in his hand rebukes the people - a stick of rulers."
Interestingly though as far as the function of hitting, the Malbim points out a crucial difference based on the Pasuk in Yeshaya 10:24:
לָכֵ֗ן כֹּֽה־אָמַ֞ר אֲדֹנָ֤י יְהוִה֙ צְבָא֔וֹת אַל־תִּירָ֥א עַמִּ֛י יֹשֵׁ֥ב צִיּ֖וֹן מֵֽאַשּׁ֑וּר בַּשֵּׁ֣בֶט יַכֶּ֔כָּה וּמַטֵּ֥הוּ יִשָּֽׂא־עָלֶ֖יךָ בְּדֶ֥רֶךְ מִצְרָֽיִם׃
Assuredly, thus said my Lord God of Hosts: “O My people that dwells in Zion, have no fear of Assyria, who beats you with a rod and wields his staff over you as did the Egyptians.
The Malbim notes that one only strikes with a 'shevet' as using such an implement is not dangerous whereas using a 'mateh' is dangerous and this is what was used when fighting the King of Cush.
I touched on this question in this essay:
The Vilna Gaon (1720-1797) writes that both a mateh and shevet are using for hitting, but differ slightly: to hit with a mateh one only needs to raise one’s hand a bit in order for the impact to be felt, while to hit with a shevet one must wind up one’s hit with energy for the impact to be felt... Rabbi Wertheimer... also explains that mateh and makel are only made from wood, while shevet can be made from wood or metal (see Psalms 2:9 which refers to an iron shevet)...
Citing an explanation from Rabbi Shmuel Dovid Luzzatto (1800-1865), Rabbi Yaakov Tzvi Mecklenburg (1785-1865) writes that the words shevet and mateh actually have different meanings, but because they are conceptually related they became interchangeable. He explains that the word shevet originally referred to the leader of a tribe, while mateh originally referred to the other members of a tribe. The leader of a tribe is similar to a mast upon which a flag is mounted because all of the members of the tribe rally around the leader. In this way the word for a leader of a tribe is homonymous with the word for stick. Since all the members of a tribe are united behind their leader, references to them can be subsumed under the word used for the leader. Thus, the word shevet also came to mean members of a tribe because the leader of the tribe embodies the entire tribe itself. In terms of sticks, he argues that shevet and mateh can both refer to the exact same type of stick, but they refer to different parts of the stick. The word shevet refers to the top of the stick (just as the leader sits atop the hierarchal structure of a tribe), while the word mateh — seemingly related to the Hebrew word lematah (underneath) — refers to its bottom (just as the other members of the tribe live under the leadership of their clan chief).
Malbim (to Gen. 49:28) writes that the word shevet does not literally mean “stick”, rather it means “branch,” which was the most common item used as a stick. Based on this, he explains that shevet means branch and tribe because each of the Tribes of Israel is simply a branch of the greater family tree of Jacob’s descendants.
SOURCE: What's in a Word?, "Tribesmen Stick Together"