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Do Interrogator Chaplains have a different model to Chaplains (excluding named ICs)? I'm trying to find one but can't seem to.. if not, how are they differentiated (if they indeed are)?

They are the same.

I believe the only models specifically being Interrogator Chaplains are Asmodai and the one in limited edition DV?

No reason any Chaplain figure couldnt represent an Interrogator Chapplain, but if you include both in your army, it might be wise to model some sort of difference. Notably Asmodai and the one in DV both had full cloaks.

Fluff wise an Interrogator Chappy is a higher ranking member of the inner circle, and performs a different set of duties to the regular variety who are the usual spiritual religiousy kind of characters. IC's job is to perform the interrogation of captured fallen, make them repent and then kill them, extracting any useful info that might be possible along the way.
The cloak is probably an essential part of the IC's get up considering status within the inner circle.