The definition of a tropical depression on the NHC website is as follows:

A tropical cyclone in which the maximum sustained surface wind speed (using the U.S. 1-minute average) is 33 kt (38 mph or 62 km/hr) or less.

Meanwhile, a tropical cyclone is defined as follows:

A warm-core non-frontal synoptic-scale cyclone, originating over tropical or subtropical waters, with organized deep convection and a closed surface wind circulation about a well-defined center.

So thunderstorms (organized convection) are a criteria for a tropical depression because of the tropical cyclone criteria. This may not always be consistently applied, but in the case of former T.D. 10, I think it is more of a case that the system simply wasn't strong enough to be classified. While it isn't listed on the website, in practice there seems to be a lower limit of 30 mph sustained winds in order for a system to be considered a depression. Since T.D. 10 peaked at 35 mph (estimated, obviously) and then weakened, it appears to have fallen into the unofficial category of "remnant low".
organized deep convection.. lets see. the system has been more or less maintaining 1.0-1.5 dvorak ratings, implying some degree of organization. considering that the convection pattern has been sheared, it's probably been underrating the actual low-level structure... they've been getting 25-30kt satellite wind vectors to go along with that. after watching irene charge through days of shear with an exposed center, the nhc declares dead a remarkably similar system. it just isn't consistent.