Buddy: bud·dy /ˈbʌdi/ Pronunciation Key - Show Spelled Pronunciation[buhd-ee] Pronunciation Key - Show IPA Pronunciation noun, plural -dies, verb, -died, -dy·ing. Informal.
noun 1. comrade or chum (often used as a term of address).
2. bud2.
verb (used without object) 3. to be a companion; be friendly or on intimate terms.
Verb phrases4. buddy up, a. to become friendly; be on friendly or intimate terms.
b. to work closely together
Synonyms: associate, chum, co-worker, comate, companion, comrade, confidant, crony, fellow, intimate, mate, pal, peer, sidekick
Partner: part·ner /ˈpɑrtnər/ Pronunciation Key - Show Spelled Pronunciation[pahrt-ner] Pronunciation Key - Show IPA Pronunciation
noun 1. a person who shares or is associated with another in some action or endeavor; sharer; associate.
2. Law. a. a person associated with another or others as a principal or a contributor of capital in a business or a joint venture, usually sharing its risks and profits.
b. special partner.
3. silent partner.
4. a husband or a wife; spouse.
5. either of two people who dance together: my favorite partner in the waltz.
6. a player on the same side or team as another: My tennis partner was an excellent player.
7. partners, Nautical. a framework of timber round a hole in a ship's deck, to support a mast, capstan, pump, etc.
Synonyms: accomplice, ally, assistant, associate, bedfellow, buddy, chum*, co-worker, cohort, collaborator, colleague, companion, comrade, confederate, confrere, consort, crony*, date, friend, helper, helpmate, husband, mate, pal*, pard, participant, playmate, sidekick*, spouse, teammate, wife
After reading all that, I don't see a real advantage of one over the other. They both accurately describe the relationship. OTOH, if it ain't broke, don't fix it. The other point is the term "buddy" is so deeply part of diving that it would be impossible to change even if it were inappropriate. Looking at your profile, I see you are pretty new to diving. Give it some time, it'll grow on you.