First off all, there are three different characters.
這 - zhe4 zhei4
那 - na4 nei4
哪 - na3 nei3.
As for the difference between the e and ei sounds. A friend of my from Taiwan told me that it's simply people being lazy and slightly slurring the sounds in 這一個. This sounds like zhei-ge when you say it quickly and don't enunciate clearly. This is true for 那 and 哪, to.
A friend of mine from Beijing told me that everybody in Beijing always says zhei and nei, and actually they never say zhe and na over there, it's like a local thing. After she told me this, I have listened very carefully in Chinese class to two of my older teachers (both of whom are from Beijing) to see if what she said is true or not (I have been doing so for at least 10 weeks and have 4 hours of class a week). Both of them always say zhei and nei, and I haven't yet heard either of them say zhe or na.
One of my other Chinese teachers, another really authentic Beijinger, told me that the difference is a case of "formal and informal". I'm currently preparing for a speaking competition and he said that I ought to use slightly more formal language. When I said ”那個是哪一個城市呢？ he told me off, saying it'd be better to say na3. When he spoke for the next 10 minutes or so I really paid attention to what he said to see what he would say when not really paying attention to his own speech (which is the best way to hear what people actually say). He mostly said zhei and nei, but he did say na a couple of times too. It sounded weird, it didn't really fit. Maybe it really is a Beijing thing.
I have a friend from Shanghai who never says zhei and nei, to him it's always zhe and na. (Which makes me think that it is a Beijing thing).
So I don't really have an answer for you, just what people have told me and their ideas.
There is no difference in meaning, so I suppose you can say whatever you like really.