The three can can be roughly split by their schooling or status within the mage-clans.
Magi can be applied as a generic term or, more properly, to one of the mage-clans of the East.
Sorcerors do not have that official affiliation, but share the magi's training and use magic in a similar way. They can be thought of wandering magi.
Wizards are rogue magic users, born of their own talent rather than schooling within the mage-clans and their schools of magic. They do not respect the same norms as the magi and sorcerors.
From the Diablo I manual:
Although practitioners of the mystic arts are scarce within the often
superstitious and religious lands of the West, many magi have made the
pilgrimage from the Far East to see for themselves what horrors lie
beneath the ruined Cathedral of Khanduras. The veiled Brotherhood of
the Vizjerei, one of the eldest and most dominant mage-clans of the
East, has sent many of its acolytes to observe the dark events
unfolding in Khanduras first hand.
The Vizjerei, known for their brightly colored turinash -or
spirit-robes have taken a keen interest in both gathering knowledge of
demons and seeing them slain. The Vizjerei elders hope that their
acolytes will learn the secrets of the dark evil that they sense
growing in the West and can destroy it. The possibility of discovering
long-lost tomes of magical knowledge within the confines of the
labyrinth has also captured the interest of many wandering Sorcerers.
From Blizzard's class description of the Wizard in Diablo III:
Magic-users hail from academies throughout Sanctuary – from Xiansai to
Caldeum – bearing monikers like "sorcerer" and "mage," but those who
would refer to themselves with the derogatory epithet "wizard" are as
similar to their fellow spellcasters as a lion is to a kitten. Wizards
and sorcerers both wield the hidden mysteries of the arcane; there,
the commonalities end.
Wizards are known for a number of qualities: not only rebelliousness
and flair, but also disdain for the endless lessons and prattling
about caution and safety that echo from academic schools of magic.
Wizards’ superior attitudes seem to stem from their natural talent –
their ability to wrestle the ambient force of magic into submission
and direct it to their ends by will and instinct. Any accidents that
might occur due to their lack of finesse are unfortunate…but that
rarely stops wizards from indulging in their unstable power.