As I see it right now they both mean: to carve; to engrave, to chisel; to notch.
Is there any difference between them?
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Sign up to join this community「[掘]{ほ}る」 actually does not mean "to carve", "to engrave" or "to chisel". Instead, it means "to dig (up)".
「[穴]{あな}を掘る」 = "to dig a hole"
「いもを掘る」 = "to dig up potatoes"
You are probably thinking of 「[彫]{ほ}る」, which means what you stated.
「メダルに[名前]{なま}を彫る」 = "to engrave one's name on a medal"
「[仏像]{ぶつぞう}を彫る」 = "to carve a Buddha"
Traditionally, 「彫る」 has another important meaning which is "to tattoo".
「[刻]{きざ}む」 can mean "to carve", "to engrave", etc., but its primary meaning is "to cut into pieces", "to chop", etc.
「刻みキャベツ」 = "shredded cabbage"