Both mean "lonely" and appear to be valid readings for 寂しい.
Is there a difference in nuance? Is this difference due to dialect?
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According to an answer to a similar question on Goo's oshiete site:
「さびしい」 is generally used in two ways (roughly equivalent to how the english "lonely" is used):
These two meanings are essentially subjective and objective, respectively, and さびしい can be used in either. さみしい, however, generally tends much more toward the subjective/emotional meaning of (1).
As for whether it is dialectal, I do not believe that it is. The answer to the question linked above suggests that this b→m sound change is not limited to さびしい／さみしい alone, and cites other examples such as 煙る（けむる／けぶる）and [目を]つぶる／つむる.
Update: Inspired by @Axioplase's comment below, I investigated the b/m sound change, specifically regarding 「寒い」.
Wiktionary suggests that in fact, the origin of さむい is the same as さびしい: さぶし. At some point this split into さびし (which later became さびしい and subsequently さみしい), while さぶし also went on to form さぶい -- and subsequently, the modern さむい which we use today.
Unfortunately, I can't find many other (reputable) sources for the etymology of さむい.