Verbs or predicates are at the end of the sentence, therefore question sentences usually go just before them. In this sentence, although it's not very clear, the predicate is ben. A better-constructed translation to English would be "Why is it me?" and there is a hidden (invisible) 'to be' verb within the pronoun ben. :)
They are essentially interchangeable as the English word "why" :-)
Technically speaking, there is a VERY subtle difference. You don't need to know this, but if you're curious:
- "neden" = literally "from what" (as in, "what caused/inspired this?")
- "niçin" = literally "for what" (as in "what is the end goal/purpose?")
- "niye" = literally "to what", and has the exact same meaning as "niçin"
I know you didn't ask about "niye" but it's quite popular, as well. :-)
Strictly speaking, "Ben" (nominative case) = "I"; and "Beni" (accusative case) = "me".
In practical use, however, "ben" usually translates as "me" when it is the predicate (i.e. completes the sentence in a "to be" or "to not be" statement) For example: This is me. Why (is it) me? That wasn't me.
Turkish is very consistent, so this irregularity stems from English: when the pronoun is the predicate, we use the objective pronoun 99% of the time... In theory, we should use the predicate nominatives: "I, she, he, we"... But they are so rarely used that they sound completely incorrect... to the extent that Duo probably won't accept them, haha! (For example: "Why I?" sounds terribly wrong, but strictly speaking, is grammatically correct.)
Here is a link if anyone would like to learn more: http://www.dailygrammar.com/Lesson-137-Pronouns.htm