Generally speaking "tasukeru" is for more serious rescues and "tetsudau" is more for being helpful/handy in a situation. One colleague helping another finish a project is "tetsudau." Saving a baby from a burning building is "tasukeru." Helping clean the house is "tetsudau," but driving 3 hours out of your way for someone is "tasukeru." You could also say that "tasukeru" is something where the person in trouble cannot aid himself, but "tetsudau" is where you help speed him along with something he's already doing. It isn't strictly for animates either, as a type of medicine can "tasukeru" a medical condition, but a natural disaster can "tetsudau" an economic one to make it worse.
There is some overlap, but these are pretty steady guidelines. Often you'll see learners say "tasukete kudasai" when asking for help with their Japanese, which I can't help but find amusing.
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