I think (and, again, I'm absolutely not an expert) that a fundamental difference between the two is that from our perspective gravity "effects" particles with 0 mass, while the electromagnetic field does not effect particles with 0 charge.
So a photon, which has 0 mass is still bent by gravity. Everything that we've observed that moves through spacetime is bent by gravity. That's why we say that gravity is a warping of spacetime itself, where electromagnetism isn't.
If you tried to build the "general electromagnetic theory of relativity", then 0 charged particles wouldn't follow a straight line on a geodesic of spacetime. With gravity, everything follows a straight-line on the geodesic of a curved straight line, regardless of its mass.
As to why such a difference exists between gravity and electromagnetism, that's well above my pay grade.
So a photon, which has 0 mass is still bent by gravity. Everything that we've observed that moves through spacetime is bent by gravity. That's why we say that gravity is a warping of spacetime itself, where electromagnetism isn't.
If you tried to build the "general electromagnetic theory of relativity", then 0 charged particles wouldn't follow a straight line on a geodesic of spacetime. With gravity, everything follows a straight-line on the geodesic of a curved straight line, regardless of its mass.
As to why such a difference exists between gravity and electromagnetism, that's well above my pay grade.