Yes they do have different meanings, though not consistently. Joseph Fielding Smith declared about the fall:
I never speak of the part Eve took in this fall as a sin, nor do I accuse Adam of a sin. … This was a transgression of the law, but not a sin … for it was something that Adam and Eve had to do!
-Joseph Fielding Smith, Doctrines of Salvation, comp. Bruce R. McConkie, 3 vols., Salt Lake City: Bookcraft, 1954–56, 1:114–15
Dallin H Oaks, referring to the above quote, said:
This suggested contrast between a sin and a transgression reminds us of the careful wording in the second article of faith: “We believe that men will be punished for their own sins, and not for Adam’s transgression”. It also echoes a familiar distinction in the law. Some acts, like murder, are crimes because they are inherently wrong. Other acts, like operating without a license, are crimes only because they are legally prohibited. Under these distinctions, the act that produced the Fall was not a sin—inherently wrong—but a transgression—wrong because it was formally prohibited. These words are not always used to denote something different, but this distinction seems meaningful in the circumstances of the Fall.
Further, It has been said that Adam and Eve merely transgressed the law because they did not yet know the difference between good and evil, and thus they were not accountable for sin. A similar argument is used to say that children cannot sin; they are not accountable. They do, however, trangress laws.
The distinction between the two terms is not always clear cut though and invariably the two will be used interchangeably despite the subtle difference.