Well, the most obvious difference is that the two "better" restaurants you mention serve brick-oven style pizza, which is probably quite different from the cooking method in a "cheap sub-style pizza place." Brick oven pizzas are generally cooked at a much higher temperature for a much shorter time, and the surface they are cooked on causes significant changes to the crust. You'll get much greater oven spring, a crisper outer crust layer, a dryer bottom crust, and a softer interior due to the oven spring and short bake.
All those things already will make a huge difference in crust characteristics. But the dough you're describing (rubbery, spongy, few bubbles) also sounds a bit underdeveloped. Higher-quality pizza shops will often allow a longer time for fermentation (which also increases flavor, as well as producing better texture); the stuff you describe may be "rushed" somewhat.
Although you say "thin crust," I'm assuming you're not talking about the thinness of, say, NY Style, but rather an intermediate crust depth (not as much as "Sicilian" but not as thin as brick oven, Neapolitan, or NY Styles). At least that's the kind of crust I associate with the kind of "sub-shop" pizza you're describing.
If that's correct, there could be other problems too. Places like that may not have confident workers stretching the dough, and they overcompensate to avoid tears or holes -- this means they often don't stretch the dough as thinly, may roll out the dough and deflate it a bit, may use a lower hydration dough (which is easier to handle), and may use a higher-gluten flour. Lower hydration doughs can often produce a denser and less tender product. Moreover, high-gluten flour will toughen significantly as the pizza begins to cool, and a thicker crust improperly baked may also have a "gum line" right below the ingredients where the dough isn't cooked as thoroughly. The moisture from that gum line can also migrate down to the crust during cooling and toughen the crust. Pizza packaging can also make a difference here: those rippled sheets some places put pizza on help to take moisture away and avoid some of the toughening in heavier crusts; without them, a dense dough can also toughen.
To sum up, here are a few things that may be contributing to the "rubbery" experience compared to your preferred brick-oven style:
- higher gluten flour (when used inappropriately)
- low dough hydration
- inadequate fermentation time
- inadequate rest after shaping
- heavy rolling out of dough, rather than stretching
- lower oven temp
- less ideal baking surface
- longer bake time
- thick or medium crust not baked through
- bad packaging
There are other possible causes, but I imagine these are part of the issues you've experienced.
As for the hypothesis of frozen or pre-made dough, that could be an issue, but there's no reason pre-made frozen dough necessarily will produce bad crust. (There will generally be some quality loss, but it's mostly flavor.) It's perfectly possible to make a tender, flavorful, non-rubbery crust with frozen dough (assuming the proper recipe). Frozen dough might contribute to the problem if defrosting and proofing time is inadequate; while convenient, frozen dough needs plenty of time to thaw and get ready before baking.
That said, a final issue in some pizza places may be dough schedule. A lot of small-scale pizza shops don't make new batches of dough every day, and if they aren't careful about storage and prep, the dough that's "too young" or "too old" may have undesirable characteristics, including bad texture. (As mentioned above, your description sounds like "too young" dough, if anything.)