What is the difference between "’ll" and "will"?

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.