Answer:
There are some very good answers here, but I thought I’d add mine to flesh out a little more detail.
Both names have Irish heritage, but one is ultimately Irish in derivation, while the other traces its earliest roots to Middle English, with possible earlier links to Latin.
Dean was at times spelled different ways—such as the first recorded use: Robert de Dene, who was an official in charge of wine and other beverages for King Edward the Confessor (1042-1066).[1] According to Wikipedia[2] , it's primarily thought that Dean is what's called a 'habitational' surname--i.e., a surname that denotes where a person is from, as "dene" was a middle English word for a valley. They also note that some people believe it to be an occupationally derived surname, related to an ecclesiastical Dean, which would then derive from the Middle English word 'dene', which ultimately traces its etymology to the Latin "decanus" (which lead to the modern English word "deacon"). Many Deans (with various spellings) did end up in Ireland, but the ultimate origin is English.
TLDR #1; the origin of Dean as a surname (last name/family name) is Middle English, possibly ultimately stemming from Latin. The given (first) name is also thought to have roughly the same etymology.[3]
Sean, however, is much more purely Irish, though it was an Irish variation of the Anglo-Norman names Johan and John.[4] I think this is important, because Sean rhymes with John in modern pronunciation, and I'm guessing it did back then, too (if someone knows otherwise, please comment and correct me). Shawn and Shaun are later English spellings based on the sound of the Irish name. The Wikipedia page (Ibid) notes that in southern parts of Ireland it's commonly pronounced the way we say it in modern English (i.e., like Shawn/Shaun, rhymes with ‘dawn’), but in northern Ireland it's often "pronounced 'Shan', 'Shen' or 'Shayn'…thus leading to the variant Shane," which I thought was interesting.
TLDR #2; Sean is more distinctly Irish, and was based on John/Johan, which is why it sounds the way it does.
TLDR #3; English is weird, and it begs, borrows and steals from other languages willy-nilly, without any consistent plan for when to keep the original language’s pronunciation or spelling and when to modify it. This confuses non-native learners of English to no end. As mentioned in another answer (which I only saw after I wrote this) there's a great poem, rather appropriately titled The Chaos, by Gerard Nolst Trenité, which gives a wonderful example of the situation and is highly worth a read, even if it might tie your brain in a knot ;).
Explanation:
