99% of the time when people pronounce "a" as a bigger sound (like how you pronounce the letter itself) as in "I am a student," it's to correct someone or emphasize the fact that you are one of MANY students. If someone said to me, "Are you the student?" I would say, "I am A student," to show that at least I don't think I'm any student in particular, just one of many. It's the same principle in pronouncing "the" (thuh) and "THE" (pronounced like thee). "Are you a new employee?" "No, I'm THE new employee," as in, I'm the only new employee, or possibly I'm a specific employee you need (may sound arrogant). So, there is an added connotation and function, but both pronunciations grammatically mean the same thing.

I agree with R. Miles answer, but I think your question is really concerning the simple pronunciation of the word "a" as it is used in normal speech. So, using your example of "I'm a student", you have already noticed that people pronounce it two ways. The reason for this is that it is easier (at least for US speakers) to form the "uh" sound than it is to form the "A" sound, particularly when it is coming directly after the "mm" sound of "I'm". If you practice saying both of them slowly, you'll see that the "uh" sound only requires one to open their lips a bit. What's more, their mouth is now in a pretty good position to go on to the "ss" sound in "student". On the other hand, to make the "A" sound, one has to open their lips a bit further and bring their tongue back a bit. So, in my opinion, the reason you hear people pronouncing it two different ways is that some people simply get lazy in their pronunciations.

When "A" is a noun, that is the first letter of the alphabet it is pronounced as "eɪ" When "A" is an indefinite article it is pronounced either "ə" or the strong form "eɪ" In other words, in the following example you can pronounce "A" using either of the two examples mentioned. For example: A man

actually both have the same meaning