In both the older novels and the new Thrawn, Thrawn is introduced to the Empire after a ship looking for smugglers finds the camp on a remote planet where he's been serving out his exile. Thrawn manages to get aboard the Imperial ship where the captain is so impressed that he takes him to Coruscant. The new book explains that Thrawn's exile was actually a ruse by the Chiss Ascendancy to enable them to gather intel on the Empire and ascertain whether or not it was a threat to them. The older Legends books never gave any indication that Thrawn's exile was anything other than that, and he certainly broke Chiss rules in the book Outbound Flight.
As far as personality differences, in Thrawn, he is somewhat politically naive and seems to greatly care for minimizing casualties among both sides. From what I remember, he's colder in the Legends books than he is in Thrawn. It's hard for me to say if any personality differences are due to the re-writing of his character, or if it's just because of the times when the books take place. Thrawn is set a while after Revenge of the Sith, while the old Legends books (with the exception of Outbound Flight) were set after Return of the Jedi.
Edit: The new book is just titled Thrawn, and the Legends ones I was referring to are Heir to the Empire, Dark Force Rising, The Last Command from the Thrawn Trilogy, and Outbound Flight.