What is the difference between an alphabet, an abjad and a syllabary?
They are different writing system types, distinguished by what each written symbol represents.
An alphabet has separate symbols for consonants and vowels, both written. Latin, Greek, Cyrillic and Armenian are alphabets. English uses 26 letters to represent roughly 44 sounds, which is part of why its spelling is so irregular.
An abjad represents consonants only, with vowels either omitted or marked optionally with diacritics. Arabic and Hebrew are abjads. Readers supply the vowels from context and vocabulary knowledge, which works because both languages have root-and-pattern morphology where the consonantal root carries the core meaning. Vowel marks are used in religious texts, poetry and material for learners.
An abugida (or alphasyllabary) is a fourth type worth including. Each symbol represents a consonant with an inherent default vowel, modified by diacritics to change the vowel. Devanagari, used for Hindi and Sanskrit, and most other South and Southeast Asian scripts work this way, as does Ethiopic.
A syllabary has a distinct symbol for each syllable rather than each sound. Japanese hiragana and katakana are syllabaries, with roughly 46 basic characters each. Syllabaries suit languages with simple syllable structure; they would be impractical for English, which has thousands of possible syllables. Cherokee, invented by Sequoyah in the nineteenth century, is another.
A logographic system uses symbols representing words or morphemes. Chinese characters are the main example — tens of thousands exist, with a few thousand needed for literacy. The advantage is that meaning is accessible across dialects that are mutually unintelligible when spoken; the cost is the learning burden.
Japanese uses three systems simultaneously — kanji (logographic), plus two syllabaries — which is unusual and reflects its history of borrowing.
No system is superior. Each suits the phonology of the language that developed it.