They don't really -- in particular, the picture you might have from high-school physics lessons of electrons orbiting a nucleus like planets orbit a star is really wrong. And it's not really the same forces; the way an atom looks is mostly down to the electromagnetic force, whereas the way a solar system looks is mostly because of gravity.
On the other hand, it is true that Newtonian gravity and the electrostatic force (i.e., what the electromagnetic force becomes when you make the approximation that there are no moving charges) have very similar mathematical forms, with the same sort of differential equation governing them. What Newtonian gravity and electrostatics have in common is mostly that they are convenient simplified approximations; you could say that they're so similar because there's a limited repertoire of differential equations simple enough to make good approximations. (There's a bit more to it than that; for instance, inverse square laws fall naturally out of the geometry of a 3-dimensional world.)
Indeed, some patterns in nature might look familiar. For instance, stellar matter in novae seem a lot like clouds but only because human psychology is really good at pattern recognition.
As for Newtonian physics versus relativity Brian Green’s Elegant Universe TV show (and book) does a great job illustrating the differences in an entertaining way.
No: not the same forces. The structure (and motion) of the macro-verse is governed entirely by gravity. The structure of the micro-verse is governed by the EM, weak and strong forces. Gravity, being about 10^25 times weaker than the next weakest force (the weak) has no influence at quantum scales. And as gjm11 points out below, the "solar system" model of the atom is a old classical model of convenience, hypothesised before the work of bohr, fermi, dirac et al worked out the quantum model. It's still taught in schools because the QM model requires fairly advanced math.
On the other hand, it is true that Newtonian gravity and the electrostatic force (i.e., what the electromagnetic force becomes when you make the approximation that there are no moving charges) have very similar mathematical forms, with the same sort of differential equation governing them. What Newtonian gravity and electrostatics have in common is mostly that they are convenient simplified approximations; you could say that they're so similar because there's a limited repertoire of differential equations simple enough to make good approximations. (There's a bit more to it than that; for instance, inverse square laws fall naturally out of the geometry of a 3-dimensional world.)