Sure, I can help with this!

The two words have similar meanings, and in a lot of cases, you can say either one. "This burned food is awful. This burned food is terrible." Same meaning.

However, there is a subtle difference. For one thing (probably you already know it but I'll mention it for good luck), "awfulLY" can be used just like the word "really" or like I sometimes use очень in Russian - to mean much, a lot, extreme, etc.. In that case there is no negative meaning at all. "This gourmet food is awfully good! This gourmet food is really good!" Same meaning.

The two words themselves mean almost the same thing, but here's the difference.. the root of Terrible is "Terror" and the root of "Awful" is "awe". (I'm somewhat shortening logic here, and linguists could argue this point, but for explanation purposes, it's accurate.) So, whenever anything is Terrible, it is bad to the point that someone could be terrified by it - not only bad, but INHERENTLY bad, and a lot of times this means bad beyond hope of repair.

"Awful" is nearly the same meaning, but its exact meaning is "so bad as to inspire awe". At some point in history, the "bad" connotation wasn't present, and only the "awe" was. Over time, we came to understand it as meaning "bad" too. So if something is awful, there might still be hope it can be made better.

This is really me splitting linguistic hairs, and for 99% of everything, these words mean the same thing - in general usage, no one will ever wonder why you used one word and not the other.. In order to understand the difference, I had to dissolve the word into its root parts.

Thanks for the question!