I have no official knowledge on this subject, so keep that in mind when reading this.
I would NEVER put a DeWALT lithium battery in ANY other tool. The 20v MAX battery has virtually NO electronics in it, so the chances of it being damaged by being used in a tool that was not designed for is very high. In my mind there is absolutely no point in ruining a valuable battery, that is worth about as much as the string trimmer, when you can buy another B&D battery for $30-$50. I do not imagine that this would go well. That's just my 2 cents though.
Thanks jkones, You may be right about misappplying the battery into some application with a very different current draw but I would not expect to have a problem with the battery or the tools. I am unsure if there is any sort of circuit protection in either of the batteries.
Lithium batteries in power tools always HAVE to have an electronic component to monitor a number of factors. The electronics are programed for specific batteries, so you need to use the correct batteries with the correct tools. The 20v MAX DeWALT battery has essentially zero electronics, because the 20v MAX tools have the electronics in the trigger of the tool. So if you hook the 20v MAX battery up to something that is not programmed to shut the battery down at the correct discharge amount, it will ruin the battery. The B&D will be significantly less intensive of electronics, but the same idea applies.
For you reference on the brand names, B&D is Stanley Black & Decker's consumer brand, while DeWALT is the professional/premium brand; so though both brands are held by the same company, they are VASTLY different quality level tools.
Hope this helps!