Aches and Pains
Head and ache are collocations (word partnerships) that always go together. Head/tooth/stomach/back don't collocate with pain as a compound but you can say: I have a pain in my wrist. These words collocate with ache. In addition "an ache" is a continuous dull pain whereas "a pain" can be either continuous or sudden.

Toothache/headache/backache are written as one word. Stomach ache as two words. Ache is both a verb and a noun: My back aches / I have got a backache. Pain is only a noun. Alternatively you can use "hurt" as a verb: my back hurts. Another word used as an adjective is "sore" : I have got a sore wrist or a sore throat.
Some more examples:
I have got a pain in my chest
I have got sore feet from jogging
In written English you can use "ache for" to mean "a strong desire, long for":
I was aching for home / He ached to see her.

Ironically "ache" and "pain" can collocate as in: aches and pains
Heart ache is figurative to mean worries
“Pain” can also collocate with “pleasure” to form compound nouns (bi-nominals). Pain can be: dull /sharp /stabbing. People can “double up in pain”. Pains (in plural) can also mean effort: go to (take) great pains (be at pains) to help them. Some people are a pain in the neck (get on your nerves).

Pain can be both a countable or an uncountable noun because it can be physical (body) or mental (emotional).