There are profound physical differences between gravity and magnetism. The most important is that gravity affects all things passing through - charged matter, neutral matter, light, neutrinos, and even gravitational waves are deflected, all identically, by massive objects. It seems to be a property of the spacetime passed through, commonly explained these days as curved spacetime. Einstein and all that.
Magnetic fields, being one part of the electromagnetic field, affect only charged particles and things that have electric or magnetic dipoles (or more complex structures). Light, despite being an excitation of the electromagnetic field, passes by charged objects without deflection. Compared to electric fields, magnetic fields are interesting in that it acts perpendicularly to a charged particle's velocity, and increases in strength with higher speeds. (Knowing this is the secret to doing well playing Space Science Institute's Magnetic Golf game.)
Electromagnetism depends on "charge" whatever that ultimately really is, for quantifying both the strength of a field surrounding a charged object, and the response of a test particle in that field. Gravity is different. You could propose that mass is like charge - just compare Newton's inverse square law with Coulomb's - but no, there's something different. It really is all about spacetime geometry. Mass quantifies how much curvature you get from a given lump of mass/energy/momentum, but all test particles passing through a place are deflected the same regardless of their specific natures.
There are some interesting similarities between gravity and magnetism, however. If you are on a rotating platform and toss a ball, the ball will appear deflected by a "Coriolis force" appearing to push perpendicular to the ball's velocity, and also depends on the ball's speed. Interesting, but non-profound.
Also, it should be noted that massive spinning objects such as the Earth do produce a twist in spacetime that could be compared to a magnetic field. This gravito-magnetic field bears the same relation to the ordinary radially attractive gravity field due to the mass, as the magnetic field around a spinning charged object bears to the radial Coulomb field due to the charge. We are looking at the space or the time components of a holistic 4-dimensional tensor. This is profound, at the level of math and ways of representing physical fields with tensors and vectors, but still does not indicate profound physics.