I have eaten at both in the last two weeks.
Blue Door: Funkier, longer waiting line. Smaller space. Almost like something I'd expect to find in NYC. I like the feel of the Blue Door over Bulldog NE.
Bulldog NE: Slicker space. Cooler down-town crowd at lunch.
Honestly, I liked both of them but wasn't in love with either of them. They're about the same size, both have nice quality buns. The Blue Door specializes in cheese-stuffed and Bulldog seems to have more odd (odd is good, imo) flavor combinations like garlic chunks + stilton or japanese togarishi + slaw.
Both places have good beer selections with lots of decent local brews.
The Bulldog clearly won the Tot war. They have a superior Tot. The Blue Door tots were not moist inside and didn't have great crunch on the exterior. They seemed like basic store-bought tots that had been held in a warming oven or steam table, while the Bulldog's super-tots seemed to have been deep-fried to order in properly hot oil and delivered within a few minutes of coming out of the fryer.