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We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\mathbb{Z}/n$, $n$ invertible in $k$, agree in sufficiently high degree.

Could someone please describe what is known and what is conjectured about the relation between $K_*^{et}(X)_{\mathbb{Q}}$ and $K_*(X)_{\mathbb{Q}}$?

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They are always (for any scheme) equivalent using transfer maps as in Proposition 2.14 in Thomason's paper.

For smooth qcqs schemes, more is true - the etale and Nisnevich rational motivic stable homotopy category are equivalent (this is essentially because the Nisnevich and etale cohomology coincide rationally) so the equivalence is true for general motivic spectra (like algebraic $K$-theory; in the non-smooth case you get a statement about Weibel's homotopy algebraic $K$-theory).

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