Is there any difference in the meaning when we use 'll or will?
I will go to university tomorrow.
I'll go to university tomorrow.
A major role of language is establishing a social context, and contractions are one of many usages to establish an informal context. Back when I was in university, informalisms were frowned upon, but the language is moving towards stuff you can say quickly, so maybe they are acceptable now.
There are plenty of markers of a formal context that sound unnatural to a (my) modern ear, e.g., "we are not amused" or "it is thought that". On the other hand, the third person plural and passive voice looks better in print, which is a reason for the divergence of the written and spoken language.
Incidentally, saying "go to university" to an American as opposed to "go to the university" would establish that you're English, and therefore culturally superior to an American.
I'm not sure if there is any difference in meaning in the words themselves, since one is a contraction of the other. But they tend to be used slightly differently, with the contracted form more likely to appear in spoken English, for example. And changing the intonation and context might change the meaning completely.
- I will go to university tomorrow.
- I'll go to university tomorrow.
3a. "If you'll start the cart, I'll make sure the house is locked." Another informal offer by way of a suggested arrangement.
4a "If you will start the cart, I will make sure the house is locked." - this sounds almost like a command.
You cannot use "'ll" as a short answer:
"Who's going to pay for the meal?"
Do not use "'ll" in legal or business documents. It is a contraction of both will and shall, and shall and will have different meanings in law.
It may sound archaic nowadays, but I think that the correct expansion of "I'll" is (or rather was) "I shall". For first person, the auxiliary verb for indicating future is "shall" (it is a bit more complicated actually, see for example the Wikipedia article. If you write or speak the contraction, you can "leave this question open". This is one aspect.
The other is that writing contractions makes your prose more informal. For example, "ain't" is not really proper english; it is rather the attempt to bring a speaking accent to paper. The same holds in principle for "isn't" and "can't", but those have been established for a much longer time now.