。
She has a different opinion from mine.
Literal
She [topic-は] me [from-と] different opinion [object-を] possesses.
違う ('to differ') takes its standard of comparison with と — 'different from X' is Xと違う. As an attributive ('different opinion'), it modifies 意見 directly: 私と違う意見. This relative-clause-as-modifier pattern (full clause + noun) is everywhere in Japanese — even short two-word clauses like 私と違う become relative clauses when they precede a noun. 持っている is the stative ~ている of 持つ ('to hold'), used for 'have/possess' since possession is a continuing state.