I have seen the word "heut" and searched in Wiktionary, it says it is an alternative (colloquial) form of "heute". What is the difference between "heute" and "heut"? How to use them?
There is no difference in their meaning. They mean the same. But there is a difference in the context where you can use them.
This is the "official" form of this word. You can read it in newspapers.
This is the colloquial form of the same word. You will not find it in newspapers (except for verbatim quotes of spoken phrases) but you will hear it very often when native speakers chat with each others. Maybe in some decades or centuries "heut" will be the official form (then "heute" will be the outdated form), but today it is not.
Like every living language, German is changing all the time. Some changes happen quickly, some are slow. One slow change, that is in progress since more than a century, is the elimination of unstressed -e at the end of a word.
You can see this in the dative form of many masculine and neuter nouns, where today the -e is completely vanished:
- Old (from a well known poem): Am Brunnen vor dem Tore.
- Today: Der Ball liegt vor dem Tor.
Some month ago, someone had a question about German adjectives that end in an “e” such as “leise” and “lose”, and in my answer I wrote a list of adjectives that are loosing their -e just now. You can watch this process since many decades, and it will go on for many more decades. This is the list I wrote in the other answer:
allein, bang, blöd, bös, derb, fad, feig, mild, nah, öd, rapid, sacht, spröd, stupid, träg, trüb, zäh
They all exist also with an -e at the end (alleine, bange, blöde, ...), and in some cases the form with -e is still more common the the e-less form, while for some other words the opposite is true.
But this is nothing that is typical for adjectives. It happens in all kinds of words. The ending -e is vanishing everywhere, and it is vanishing very slowly.
But there are also some words where it can't vanish, because if you leave it out, you get a different word with a different meaning. Here are some examples:
die Falle = trapp
der Fall = case
die Buche = beech (a tree)
das Buch = book
der Zeuge = witness
das Zeug = stuff
die Reise = journey
der Reis = rice
There is no change that the -e of those words will vanish as long as the other words without -e still exists.
There is no systematic difference between the full and the shortened form except that one is a syllable shorter.
Pretty much any word ending in an unstressed e can be shortened: Katze/Katz', gehe/geh', große/gross', eine/ein'... It depends on the level of formality, the need to conform to a specific meter, the speed at which someone is speaking etc. etc. whether the ending is reduced or not.
But I know of no case where the full and shortened forms constitute a minimal pair, i.e. expressions with systematically different meanings. This seems to be purely phonetic reduction. German simply hasn't yet brought the process to such extremes as e.g. French.