The word "diphthong" comes from Greek words meaning "two sounds." That pretty much tells you the difference between diphthongs and vowels.
Pure vowels have just one sound. When such a vowel is spoken, the tongue remains still.
By contrast, diphthongs have two sounds and the tongue must move while moving from one sound to the other. A diphthong starts with one vowel sound and then glides to the other. Many diphthongs are made with two letters (like "look" and "coin") but others can have only one letter (like "ride" and "fine").
Pronunciation, in any language, is crucial for good communication and understanding. Vowels and diphthongs are not unique to English speakers, and their use is always to ensure pronunciation in whichever language they are being applied. Diphthongs relate to the phonetic alphabet, and help language learners describe and adapt to different sounds.
They are particularly important, and in English, are used because there are many different English accents across the board with American English and British English being two main varieties in pronunciation, and with Australian English also having distinct pronunciations. Accordingly, second and third language learners need a basis on which to concentrate their own efforts to pronounce words without reliance on an accent or inflection they may be used to, and because their own language will have its own set of sounds applied to vowels and consonants.
The main difference between vowels and diphthongs, without being too technical, is that diphthongs are exclusively sounds whereas vowels, although they are also sounds, are also specific letters of the alphabet from which those familiar sound combinations originate.
Vowel is a single sound of a particular type - it is an open sound which is made with free passage of breath. For example the sound of "ah...' a doctor asks you to make when examining your throat. Articulation of a vowel is not accompanied by any closure or narrowing in the speech tract which would prevent the escape of the air stream through the mouth.
In English language sounds of vowels are represented by the letters a, e, i, o, u, and sometimes w and y. Sound of vowels is distinguished from that of consonants, which are made by organs of speech more or less closed.
Once we understand what is a vowel, a diphthong can simply be defined is sound produced by pronouncing two vowels as a single syllable. A diphthong has a starting point as one vowel and the tongue glides in the direction of the second vowel. The change in sound required for diphthong can also be achieved by change in shape of lips. An example of diphthong is the word kite, where the syllable after the letter k starts with sound of a as in far, and then changes in sound of e as in be. The combination of letters ei in the word height produce similar sound. The letters ou in the word sound represent diphthong with a different sound.
In English language presence of two letters representing vowels generally spell a diphthong. But there are exception to this rule. For example the "oo" in spelling of book and "ai" in spelling of paid represents singles sounds that do not require change in position of tongue or lips in between pronunciation of these sounds. Vowels like these are sometime called long vowels. For example the word mat is pronounced with a short vowel, and the word mate uses the long version of the same vowel. In some words a single letter representing vowels is used to spell a diphthong - for example, the letter i in the word light.