Sounds of the letter a

How can I know, precisely, when to differentiate the sounds of the letter a, like in: apple and vault?

This is an extremely broad question, actually, and I doubt there can be a single, definite, comprehensive answer. I will try to provide a few quick-and-dirty rules of thumb, but be aware that you will probably find exceptions to every single one of them.

Also, as Colin Fine points out, note that things are not pronounced the same everywhere. For example, can't can be pronounced as /kaːnt/, /kɑːnt/, /kænt/, and /keənt/, depending on what variety of English we are talking about. As a second example, the vowels in Mary, marry, and merry sound identical in certain dialects but not in others.

All that being said, on to the rules of thumb.

First, the basics. Here are the Wiktionary usage notes for a:

In English, the letter a usually denotes the near-open front unrounded vowel (IPA: /æ/), as in pad, the open back unrounded vowel (IPA: /ɑː/) as in father, or, followed by another vowel, the diphthong /eɪ/, as in ace.

Now on to details, which I will be more adapting than quoting from the essay "Hou tu pranownse Inglish" by Mark Rosenfelder. Note that the order of the following rules of thumb is important. As the author puts it, "to pronounce a word, you go down the list of rules, seeing if each one in turn applies, and applying it if it does".

1. aught becomes /ɔt/ (daughter, caught, taught, naught)
2. ay is pronounced /eɪ/ (day, say)
3. al is pronounced /ɔl/ before r, s, a dental stop, or final ll (also, already, wall, bald, although), and sometimes before m (e.g. in almost, almighty; but there are many exceptions)
4. alk becomes /ɔk/ (walk), except initially (alkali, /ælk/)
5. a is pronounced "long", as /eɪ/, before an intervocalic consonant (rate, bake).
6. a is pronounced "short", as /æ/, before two consonants (battle, ladder) or before a final consonant (bat, dad, ram).
7. wa is pronounced /wɒ/ (UK) or /wɑ/ (US) before t, d, n, s (want, wander, swan, Rwanda, swat, wad, wasp), and as /wɔ/ before sh or tch (wash, squash, watch).
8. There is one thing not covered by the above: an unstressed a is often reduced to a schwa, /ə/, as in addition /əˈdɪʃn̩/ or anomaly /əˈnɒməlɪ/. However, as the author puts it, "the idea here is to predict pronunciations from the spelling, and the spelling doesn't indicate the stress". Once you hear a word, you know where the stress is, but then you also know how to pronounce any a in it anyway.
9. At this point, you are more or less safe to make the following substitutions:
   • eau = /o/
   • ai = /eɪ/
   • au, aw = /ɔ/
   • ea = /iː/
   • oa = /o/
   • ua = /juə/
10. a is reduced to /ə/ before final l (final, typical); a "short" a, as in /æ/, is reduced to /ə/ before a final n (human). "These rules don't apply to monosyllables (pal, can), nor to vowels that have already been assigned a particular value by an earlier rule (e.g. meal to /miːl/ by the [previous] rule). These rules could probably be refined; they don't apply to stressed finals, but again, the orthography doesn't indicate stress."
11. The suffix -able is reduced to /əbəl/ (lovable, usable, formable)
12. Any remaining final a is pronounced as /ə/.
13. war is pronounced /wɔː(r)/, except before a vowel (war, warlock, dwarf)
14. /æ/ before a double r becomes /e/ (marry; see disclaimer above)
15. /æ/ before any other r becomes /a/ (mark, star)

Again, these "rules" are only approximations, so take them with a grain of salt (also, I hope I haven't thrown in a few typos or copy-paste mistakes).

For further reading, see these questions:

• Written English Vowels are Odd
• Pronunciation of names that end in “h”
• Sounding “æ” vs. “ɛ” in English phonology
• Why did only English undergo the Great Vowel Shift, making pronunciation stray so far from spelling?
• What words are commonly mispronounced by literate people who read them before they heard them?

Unfortunately English orthography is difficult. There are patterns (or rules), but there are sometimes conflicting patterns, and also many exceptions.

There are also further complications where pairs of words are distinguished in some dialects but not others: for example in my (UK) accent "ant" and "aunt" sound quite different (but "aunt" does not have the same vowel sound as "vault", and "aren't" sounds the same as "aunt") but in some American accents "ant" and "aunt" sound the same.

So it is difficult to give you a clear answer. "Au" is certainly a different environment from "a", and in most cases will be pronounced differently.