I'm sure. "Darf" and "dürfte" are most definitely different. If you were to say, "May I/Am I allowed to read it?" it would be, "Darf ich es lesen?" "Would I be allowed to read it?" is "Dürfte Ich es lesen?" I mean, I guess it's possible that they both translate to may, but they'd at least mean something different. "Darf" is simply present tense while "dürfte" is conditional.
Same with "soll" and "sollte". They are both translated to "should," but they aren't used under the same circumstances. They have slightly different meanings. "Soll" is "should," as in an obligation. "Sollte" is more of a suggestion.
Hope this clarified things a bit.
The polite form would be: "May I read it" and is used frequently. I am not sure how one would differentiate in German between the polite request and a genuine enquiry asking permission to do something. "Could I read it" is frequent but is not precise in English, it may also mean "Am I capable of reading it", as when someone learning German for just three weeks asks if he could read Kant in German.
Would " Darf ich das lesen?" be correct? What is the difference between "darf" and "dürfte" in this context?
Two sentences before in this exercise, I translated "dürfte" as "could, and was marked wrong, with correct translation given being "should." In this sentence, therefore, I used "should" and was marked wrong, with the correct translation presented as "could," in the exercise. Why was "could" marked wrong before?
may I read it...that's it...not, would I be allowed to read it...obviously we are learners here...why put some meaningless half translation here to trip people up...Would I be allowed to read it, if...something, then it may make more sense.