What would have happened if Frodo had left when Gandalf told him?

[Manwe Sulimo]

The Fellowship would've been very different. Every person (other than the hobbits) who was originally in the Fellowship would not have been: Gandalf was imprisoned in Orthanc, Aragorn was hunting Gollum, Legolas and Gimli were in Mirkwood and Erebor, respectively. If Elrond was "hasty" and sent the new Fellowship out early, they would've been ensnared by Saruman trying to pass through the Gap of Rohan, having no warning of his treachery from Gandalf. All the Elves (and Dúnedain) of the new Fellowship would've been slaughtered, and the Ring taken by Saruman. This would have resulted in a war between Isengard and Mordor, with Rohan and Gondor destroyed in the middle. Eventually, Saruman would've won (because he had the Ring), and all the Free People of Middle-earth would've been enslaved.

Sounds like a good "what if" fanfiction.

There's an interesting essay about the timing of the Fellowship's actions relative to the War in Unfinished Tales, describing how if the Fellowship hadn't split, or had been a few days earlier or later, the forces of evil would have won the War.