It's a bit tricky in fantasy, but from the scientific point of view, species refers to a group of living organisms that can interbreed. Races are subspecies within the same species.
Take humans, for instance. We're all homosapiens, but within our species, there are many different races. In general and in the simplest terms we use, there are Black, Asian, White, etc. As you can see, it's more of a social construct than biology, even though there are differences in characteristics involved.
With that said, when it comes to fantasy, if elves and humans can interbreed and have children—half-elves—they're considered to be from the same species, scientific wise, but from different races. Same goes for beastmen and dwarves. Even the monsters like goblins and orcs are considered to be from the same species as humans if the requirement is met.
To conclude, humanoid living organisms are all from the same species but can be from the different races. So saying elves are a different race from humans is actually accurate, scientific wise. Species usually refers to beasts and non-humanoid creatures.