Once I decide between Saiga and Vepr, I guess the next question to ask is are there different generations/models/brands/countries-of-origin to avoid that have problems?

Rifle or shotgun?  If rifle, what caliber?



All Saigas and Veprs originate from Russia.

I have a saiga in 5.45 and a saiga 12; a vepr in 7.62x54r and a vepr 12.

The veprs are the best made AKs I have handled. Saiga is good, vepr is better.

 
Pretty much this from my understanding

True.
Vepr is heavier, slightly more refined, but most parts tend to be easier to find for Saigas.

Vepr seems to be more heavier dutyer.

I preferred the Saiga 12 over the Vepr 12, though.

What models are you comparing?

As a Saiga 12 owner who desperately wishes he had a VEPR magwell, I'd love to hear your thoughts on why you hold this opinion.

In regards to the shotguns, I own one Saiga 12 (converted) and two Vepr 12s (fixed stock and folder).
I do like the adjustable gas system on my Saiga 12 (MD Arms V-Plug in mine).  The gas system on the Vepr 12 is not adjustable from the factory (it is WFO 24/7).  The non-adjustable nature of the Vepr's gas system is not an issue in the real world, and I consider the ability to adjust the gas on my Saiga to be a minor advantage with some loads on the very heavy or very light side, but it does bear mentioning.
Other than that minor difference, the Vepr is superior to the Saiga in every way.  Fit, finish, features, and function all favor the Vepr 12.
If you really want a Saiga 12 for whatever reason, look at the 030 version.  It has nearly all of the features and advantages of the Vepr 12 (with some minor differences), and is every bit the equal to the Vepr 12 IMO (the Vepr will still have a better finish and look prettier).  You will pay good money for those feature however.  See the 030 HERE.
I will not sell my Saiga 12, I love to take it hunting, but my Vepr 12s are a big step up from it in nearly every regard IMO.






 

Double tap.





 

The flow of the Vepr hasn't been impeded like the flow of the Saiga in country. Food for thought.

Please expound?

Saigas (Izhmash manufactured) are currently blocked from import by an EO, has to do with the on-going situation in the Ukraine.  No such sanctions have been placed on Veprs (Molot manufactured), they can still be imported into the US.



For whatever reason, Molot didn't get hit by the sanctions, Izhmash did.