For example what is the difference between /aɪ/ and /a.ɪ/ or between /au̯/ and /a.u/? how they be distinguished from each other?
The difference is that aʊ is a sequence within one syllable, and a.ʊ is a sequence in two syllables. In the domain of one-syllable representations, where are two sub-variants: long diphthong ("acts like two vowels") and short diphthongs ("acts like a single vowel"). The appropriate question then is, how does the analyst decide which is the correct analysis?
These issues are decided based on analogical reasoning. The long / short question may be resolved by reference to contrastive monophthongal length, indeed it is a classic argument for the two timing slot theory of long vowels that long vowels and diphthongs behave the same way for processes that count the number of vowels (stress, tone assignment, syllable structure). Somethings vowel sequences behave like short vowels, in which case one posits a mono-syllabic mono-moraic analysis (examples: North Saami short dipthongs, Anii diphthongs in general).
Diagnosing the difference between monosyllabic bimoraic [au] and bisyllable bimoraic [a.u] requires having a diagnostic that clearly identifies syllables rather than moras (timing slots), and that is very hard to come by – this is why the prosodic analysis of diphthongs is so slippery. There are a few languages such as Kikamba which present a surface contrast between long vowels and bisyllabic (identical) vowel sequences (the realization is a durational difference), but usually, there is no contrast. One generally looks for evidence of the type "acts as a unit" or "acts as two units". It has proven notoriously difficult to establish beyond a reasonable doubt that any language has syllables (which is why Strict CV theory is a viable theory). Generally, analysts (not Strict CV participants) assume that the syllable is a universally-available unit, which means that you always have the option of [VV] versus [V.V]. In that theoretical context, the decision is usually made by reference to some other structural claim, such as "In this language, all syllables have an onset". If that is so but there are also two-vowel sequences, then they must be monosyllabic diphth.ngs.
You can think of a diphthong as being like a vector. It starts at a given point and immediately starts changing in quality till it reaches (or doesn't quite reach) its endpoint. The two symbols used to represent it can be thought of as beginning and end points of the vowel. The fact that one of the two symbols used has a short length mark reflects the fact that diphthongs normally increase or decrease in intensity.
Two consecutive vowels, in theory, are two separate monophthongs that don't change in quality. However, in realty there will often be some change in vowel quality during the production of a notional monophthong.
The difference phonologically is that a diphthong is a single phoneme whereas two monophthongs is two phonemes!