Why the dihedral group
Every mod 10 scheme built on addition has the same blind spot: addition is commutative, so swapping two digits with the same weight changes nothing. Luhn inherits this and misses the 09 to 90 swap.
Verhoeff replaces addition with the group operation of the dihedral group of order 10, which is not commutative, and applies a position-dependent permutation before each step. Order starts to matter, and transpositions stop slipping through.
Verhoeff or Damm
Verhoeff needs three lookup tables: a 10 by 10 multiplication table, an 8 by 10 permutation table, and a 10-entry inverse table. Damm reaches the same detection strength with one 10 by 10 table and no inverse step, which makes it the easier one to implement without a mistake.
Pick either when you are designing an identifier people will type. Neither is a substitute for a hash or a signature, because both are public arithmetic that anyone can reproduce.