The colours used in the Anniversary Edition are:
US: medium green
UK: pale tan, a bit lighter than the normal British medium tan
Soviet Union: dark wine purple
Germany: dark grey
Japan: burnt orange
Italy: medium-dark brown
China (infantry only): lime green, somewhat different from the shade used for the British pieces in the Revised Edition
As far as the colour differences between the two editions go, the Revised Edition uses the same colours and shades as the Anniversary Edition for the US, Soviet Union and Japan. Germany uses the same or nearly the same shade of dark grey. Great Britain is the incompatible country: the pieces are pale tan in the Anniversary Edition and a kind of pale lime green in the Revised Edition.
In the Anniversary Edition, the five main countries nearly all use the same sculpts as the Revised Edition, except for Britain (which has new Matilda tanks) and Germany (which uses the new Panther tank design from Battle of the Bulge and the Me-109G fighter from the same game). In the Revised Edition, Britain uses Sherman tanks, while Germany uses the original-version sculpt of the Panther (it has a shorter gun and the treads don’t have square side panels) and a Stuka rather than an Me-109G as its fighter (even though the Stuka was actually a dive-bomber). Some of the sculpts have small differences: for example the US Shermans have smaller turrets and the Russian T-34 are narrower. These five countries, plus Italy, also have a new cruiser piece. Apart from its distinct infantry and tank sculpts, Italy’s land and air sculpts are copied from existing Japanese ones (artillery, bombers and fighters) while its naval sculpts are copied from existing German ones (carriers, battleships, cruisers, destroyers and transports).