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.
No. The second form is a contraction of the first. Generally, contractions aren't appropriate for more formal writing (but as always, consider your target audience when writing).
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.
There are differences between 'll and will, and there are occasions in English when you cannot make contractions, but there's no great difference in the examples you've given. A very slight difference of mood, perhaps.
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.
There's no difference in the meaning as written. However, when speaking, you might use "I will" - with the emphasis on "will" - if there was any doubt as to whether the action were going to be carried out.
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.