Martins' is down the road from me and inevitably it's a pit-stop on the way home from time to time.
Drinkstore is near my work, and also gets a look-in as a result.
If I were to compare the two as bricks and mortar shops?
Drinkstore is better curated and probably has a better selection *overall*. I agree they have a lot of stuff out the back that you can't see sometimes... But if you're going in to see what's there it's kind of hard to know what's out there unless you have a name in your head. In that respect Martins' is bigger and better laid out - large american-style coolers you can browse.
Customer-service wise Drinkstore is friendlier and more knowledgable about beer, although Martins' isn't bad by most standards.
For me what edges it between the two is the Drinkstore 10% discount on Irish beer. It's a significant saving if you buy Irish most of the time, and Martins' loyalty card ends up being a bit rubbish in comparison, even if you stick with it.