How are gravitational and electromagnetic force different?

Answer 1

There are quite some differences. The most obvious one is that electromagnetism acts on charges, while gravity acts on energy (remember energy and mass are the same thing in relativity). There are also two types of electric charges, positive and negative, while there is just one gravitational 'charge'. The gravitational force is also much much weaker than the electromagnetic force; a common fridge magnet can overcome the gravitational attraction of the entire Earth!

Gravity also allows for a geometric explanation involving curves and warps in spacetime whilst electromagnetism does not.

There are also more complicated differences. For example the electromagnetic force is mediated by a spin-1 particle called the photon. Gravity on the other hand is expected to be mediated by a spin-2 particle called a graviton (I say expected because the graviton has not yet been observed, mainly because gravity is so weak).

On a more technical note forces in quantum gravity are related to symmetries of nature. The symmetry group of the electromagnetic force is U(1) whilst the one of gravity is SO(3,1).

Even more technical: because U(1) is an abelian group photons are themselves chargeless, but SO(3,1) is not an abelian group so the graviton can couple to itself (i.e. self-interaction). This is reasonable because the graviton carries energy, and gravity couples to energy.

Answer 2

Gravitational force is a weak, long-range force that acts between all objects with mass, while electromagnetic force is a much stronger force that acts between charged particles. Gravitational force is always attractive and depends on mass, while electromagnetic force can be attractive or repulsive and depends on the electric charge of the particles.