Fingerprints and key IDs are related: short and long key IDs just form the lower 8 respective 16 hex digits (32 respective 64 bites) of the fingerprint.
An example using my own fingerprint:
fingerprint: 0D69 E11F 12BD BA07 7B37 26AB 4E1F 799A A4FF 2279
long id: 4E1F 799A A4FF 2279
short id: A4FF 2279
Shorter identifiers imply larger chance of collisions (malicious or by chance). Never use short key IDs, at least rely on long key IDs. If providing the full fingerprint is not an issue, share the full fingerprint. Short key IDs can be collided within seconds using graphic cards. The long key ID is considered save for now, but only unless somebody figures out more advanced attacks: there is not much buffer left that prevents brute-force attacks. The space of possible fingerprints is vast, and collision attacks are not feasible.