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We know, the the vector form of Maxwell equations
beginalign
vecnablacdotvecE &= 4pirho labelDiff I\
vecnablatimesvecB &= dfrac4pic vecj+dfrac1cdfracpartialvecEpartial t labelDiff IV\
vecnablatimesvecE &= -dfrac1cdfracpartialvecBpartial t labelDiff III\
vecnablacdotvecB &= 0 labelDiff II
endalign

The last two of them allow us to introduce the potentials:
beginalign
vecE &= -frac1c fracpartial vecApartial t - vecnablaphi\
vecB &= vecnablatimesvec A
endalign
which tells us about gauge invariance of equations.

All four of Maxwell's equations can be written compactly as

beginalign
partial_muF^munu &= frac4picj^nu tag1\
partial_[muF_alphabeta] &= 0;. tag2
endalign

And according to the last one equation, the first one we can rewrite (use preferred gauge) in form:
beginequation
Box A^mu = -frac4pic j^mu
endequation

Now we consider the Einstein GR equations:
beginequation
R_munu = 8pi G (T_munu - frac12g_munuT).
endequation

Or in "$Gamma-$field" form (indexes are omitted):
beginequation
partial Gamma - partial Gamma + GammaGamma - GammaGamma = 8pi G (T_munu - frac12g_munuT).
endequation

We know, in weak field limit the equations get simply form externally similar to Maxwell ones (so called gravitomagnetism) in 3-vector form, or in covariant form:
beginequation
Box h_munu = -16pi G (T_munu - frac12eta_munuT)
endequation

Thus, the question: Why for covariant form of Maxwell equations we need two different entities of equations, but for the GR the only one? Or another words, can we write the Einstein GR equations for weak field limit similar to Maxwell equations in field form, not via potentials?

Scheme of theories comparison

If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_munu$ of gauge covariant derivatives is analogous to the commutator $R_munurhosigma$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_mu[nurhosigma]=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.

1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.




2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.




3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.