In practice, it is generally thought that Mage and Sorcerer are simply different Japanese-style and Western-style terms for the same thing. However, since this is Japanese-style Western fantasy, perhaps we can split the concept:
Mage: a being that commands magic.
Sorcerer: a being born already knowing how to wield magic.
Especially after Ryne Lynne, magic became widespread. Compared to Mages, who can be found everywhere in the broad sense, Sorcerers are extremely rare. Instinctive mastery of magic is a very uncommon talent. However, among Sorcerers, there are almost no individuals who have polished their magical skills to the extreme the way ArchMages have. This is because, for them, magic is simply something that has always existed as a matter of course since birth.
Regarding Mana: a Mage's Mana relies mainly on post-training development and can grow continuously. A Sorcerer's Mana, in contrast, is essentially fixed — the Mana Capacity they possess is an innate talent. Even so, the Mana Capacity that an ordinary Sorcerer innately possesses already surpasses what the vast majority of Mages could ever attain in their entire lifetime.
A simple way to distinguish Mage from Sorcerer: Sorcerers develop the ability to control Mana without guidance before reaching adulthood. Moreover, because this ability operates through a different mechanism than that of Mages, those with sensitivity to Mana can typically distinguish between the two with ease.
Additionally, Sorcerers generally possess a certain distinctive magic that is equally innate and cannot be learned by others — the Innate Magic. Most Sorcerers come to recognize their Innate Magic once they develop preliminary control over Mana.