No, there are differences in 2-cycle oils. TCW-III type works best on low-RPM 2-cycle engines such as marine outboards. Sounds like you bought the "right" stuff. Now, that said, oil types is kind of a religious argument and you'll find people who've run TCW-III in their brushcutters for years with no ill effects.
How old is the blower? Did you thoroughly mix the oil+gas mixture before adding to the tank? Are you letting the engine warm up? (this might not be necessary, depending on the engine...I've never played with a Weedeater brand).
If it runs only with the choke partially-engaged, it sounds like a fuel delivery problem. Could be as simple as a clogged fuel filter in the gas tank. Check that first since it's trivial to replace.
If it's not the filter, it's time to start looking at the carburetor. Could be a partially-clogged jet. Does it run for a few seconds with the choke wide open after you pump the primer a few times? If so, it could be a problem with the needle (maybe a stuck spring?). Depending on how much you paid for your blower, it may be worth your time to take the carburetor apart and clean it. If you've never done this sort of thing before, figure it'll take a couple hours. 2-cycle carbs for weedeaters and blowers are really simple beasts, typically only a couple moving parts and a tiny spring but the orifices are so small that they can clog if you don't winterize properly.
To do it right, you'll need to head to your local lawn/garden shop and buy the correct "carb rebuild kit" for your model (usually less than $10) so you can install new gaskets when you reassemble. If you're careful, you can reuse your old gaskets if they're in good shape. Having some sort of workbench or table and good lighting will help immensely.
/finally got around to rebuilding the carb for his Stihl brushcutter last weekend
