Hi Amanda, good question indeed and deserves a good answer. I think my english and knowledge isnt able for that. But I can try to make a humble contribution.
Tanrı is a very old (ancient) turkish word (coming from tengri) for "God or Goddess".
It means a supernaturel power in a large sens. For exemple when we say "god of vine" "god of sky" etc. we use tanrı : şarap tanrısı, gökyüzü tanrısı etc.
In philosophic or teologic explanations too its usually prefered the term of tanrı Ex: "concept of god in different religions and beliefs" "Farklı din ve inanışlarda "tanrı" anlayışı"
(We say "ilah"(arabic word for "god") in Turkish too.)
Allah refers to a unique concept of god with its islamique context. When we say Allah we mean (a unique) (god) as its described in Qoran. Muslims believes there is no god but just Allah.That's a special name. term of "god" is not so specific. In İn addition in the Quran some (99 beautiful names of Allah) names of Allah is announced too. Each one reveals some of his particularities.
The word of "Rab" in its arabic roots have some meaning relatin to "teaching" if I am not wrong. Rab educates his "kul"("abd" in arabic) ("servant" in english ?). And Muslims oftenly call their god saying "My Rab" in their prayers. There is always a close relation between Rab and his "kul".
As I've said, the question have theologic, linguistic and semantic dimensions and needs to be elaborated more adequately.
Allahu a^lem.