The missing corners on the left-hand plot are a not uncommon phenomenon when one is trying to use equalities for regions. I'm sure this has popped up before on Wolfram Community or mathematica.stackexchage.com but at the moment am not finding it.
In any case, the documentation does not show this combination RegionPlot[ImplicitRegion[,,,]...].
However, the following (even with omitting PlotRange -> Full, which is superfluous here) gives the correct graphic.
RegionPlot[0 <= x <= 5 && 0 <= y <= 10, {x, 0, 5}, {y, 0, 10},
Frame -> True, AspectRatio -> Automatic, PlotRange -> Full]