As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III,  6.1.2.1], 
$$[X/G]_n = (G^{n+1}\times X)/G$$
One has a descent spectral sequence
$$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{p+1}\times H^q(X,\mathbb{Q}))/G$$
abutting to $H^{p+q}([X/G]_\bullet, \mathbb{Q})=H^{p+q}_G(X,\mathbb{Q})$ [Hodge III, (5.2.1.1)], and this is compatible with MHS [Hodge III, (8.1.15)]. Now use the fact that the complex $E_1$ is the bar complex, which  computes group cohomology; in this case it is trivial except in degree zero. So in conclusion
$$H_G^*(X,\mathbb{Q})\cong H^*(X,\mathbb{Q})^G$$
as MHS.