AK and SVD mount rails are somewhat different. Can someone please explain the details? Are different kinds of mouts needed, or do the same mounts fit both? Some Russian scopes seem to be made AK or SVD specific, do others fit both?

The rail on the SVD is the same diameter (diameter's not really the right word -- the rails will fit all "comblock mounts") as the AK, only longer. The mount stops via a pin located on the mount up front on the mount itself. The rail is slotted to allow the pin to move forward until it hits the stop. This allows for proper eye relief on the SVD with the standard PSO scopes. Night vision scope mounts, and a few other optics ie the 3-9x "Minuta" and PKS-07 stop against the rear of the rail and are/can be used on both Drag and AK. AK mounts (brackets) usually stop at the rear of the AK type rail since the clamp is closed at the rear or in the case of early PSO's for Saiga rifles a Dragunov locking clamp had a pin added to stop the mount at the rear. The main difference in mounts is the locking clamp which is removeable from the mount .I believe the AK style clamp was developed to allow  for clearance with the rear sight leaf with the 42mm obj. bell 's on the 6x and 8x PSO's and to allow for better eye relief. Back in Afghanistan the Soviets modded 4x PSO's for some '74's by adding a hook at the rear of  the  Dragunov mount allowing it to go further fwd for better eye relief. Check out Tantal's sight for info and all kinds of cool stuff --
http://tantal.kalashnikov.guns.ru/
If I were to scope an AK I would select a 4x PSO with the Dragunov mount, remove the stop pin on the mount (unless the mounting rail is Romanian which is slotted to take a Dragunov mounting clamp -- yes I've actually put one on one), and either put a stop pin on the rail itself or rig a rear hook Afghan style. When using a Drag clamp the scope stays considerably lower and gives a better cheek weld. If eye relief is still an issue a longer butt-stock ie KVAR NATO length fixes that. 4x is IMHO optimum magnification for an AK. The Larger Obj. 6 and 8x PSO's need to use the AK style clamp in order to clear the rear sight leaf unless you take the rear sight off, which I wouldn't do.
There are weaver mounts, both high and low out there. I don't have any hands on experience with the Russian versions. I've read of a case where a Drag mount wouldn't clear the top cover on an AK but don't know anymore. Looking at pics of them, however, I'd definitely use the AK version on the AK and the Drag version on the Drag.