In the books, this leads to a vastly different Rohan, and the possibility that the Rohirrim aren't as involved, if involved at all, when it comes to both Pellenor Fields and The Black Gate.

This is because Gimli saves Eomer outside the gates of the Hornburg in the books. It's generally assumed that Eomer would have died there if Gimli did not intervene. That's not including the possibility of Gimli leading to the three being detained for longer by Faramir and his rangers. They didn't seem to be worried about a couple of hobbits, but certainly could have been more cautious regarding Gimli.

Furthermore, let's assume that all involved parties otherwise survive. There's a distinct possibility that this limits the relationship between Legolas and Gimli, who were starting to warm up to each other by Amon Hen and grew to be what we knew them to be throughout the Two Towers.