If I wanted to talk about "the death of Caesar", I wouldn't think twice about using the genitive (mors Caesaris). But if you asked me what sort of genitive this is—possessive, partitive, or objective—I wouldn't be able to tell you.
These distinctions seldom actually matter in practice, but there's one important case when they do: with personal pronouns. The possessive genitive uses the form meus/tuus -a -um, while all other genitives use the form meī/tuī.
In particular, would "my death" be mors mea, or mors meī? Equivalently, would "your death" be mors tua, or mors tuī?
This came up in the translation of a tattoo.