If you are referring to combat aircraft, most have anhedral wings, Soviet or not. The F-15, for example, has a downward angle on the wings. This is principally because the more dihedral effect present, the more stable the aircraft is, which tends to be a bad thing for combat aircraft that need to maneuver. Anhedral wings reduce stability for this purpose. Also, aircraft with swept wings tend to be much less dihedral, or even anhedral, because the wing sweep conveys some dihedral effect on its own.

Going larger, it is also true that many larger Soviet aircraft have anhedral, both civilian and military, whereas that is/was rarer in the rest. This is because many larger Soviet civilian aircraft were repurposed from military transports or bombers. For example, the Tu-154, the workhorse Soviet airliner between 1970 and the early 2000's, was one of the fastest subsonic airliners produced, with a high degree of wing sweep, owing at least partly to military design philosophies. The same can be seen in the preceeding Tu-104 aircraft, developed directly from military versions.

Finally, on larger aircraft, anhedral is usually incorporated when the wing is high on the body, and this is true in both Soviet and western designs. A high-wing aircraft often has excess dihedral stability, and anhedral is introduced to compensate. The Soviet Il-76 and An-12, and American C-5 and C-17 are very good examples of this.

In what era/what sort of aircraft? I can find you many examples of western anhedral aircraft and I can find you examples of soviet dihedral aircraft.

Edit: each serves their own purpose, it's not done for aesthetics so you can find examples of each with both the Soviets and the West.

Could someone post a quick summary (with pictures, if you think you're up to it) of the difference between these wing-types?

Is it really true that Soviet planes have anhedral wings? I was always under the assumption that Russian passenger planes have more flexible wings. This means that they look anhedral when the plane stands, but they are dihedral when the plane flies.