I often struggle with the difference between the two terms, in practice.
Technically, transliteration is concerned primarily with accurately representing the graphemes of another script, whilst transcription is concerned primarily with representing its phonemes.
In addition, transliteration is concerned with scripts, whilst transcription is concerned with writing systems (the way a script is applied to a particular language).
Transliteration is useful when studying some aspect of a language or script which does not require an intimate understanding or representation of the language’s phonetic systems, or where the pronunciation differences between languages or dialects are not important. Transcription is useful for study which has a greater focus on phonetic systems, for example when determining how closely related two languages are, for language learners wishing to improve their pronunciation, or for examining dialectal variation within a language.