Not much has changed between first and second edition.
Both editions' boxed sets include a softcover version of the rulebook, a map, dice, a GM screen, character sheets, GM scratchwork sheets, reference cards for weapons and statuses, action cards for scripting combat, and an extra book with some extra rules. This extra book, "New Rules, New Missions", is purchasable as a standalone PDF and has not changed between editions; the softcover rulebook is the same as a standalone hardcover one.
The second edition boxed set includes an extra set of dice; the first edition set includes some giant mouse-headed colored pawns instead, which are nice if you want to play Gwendolyn on the included map, but can't really help with many game functions.
The reference cards in the second edition have less text on them and more fun graphic design, if that's your bag. If you only have a PDF of the first edition and you're looking for a boxed set, get the second edition one if you can.
Rules function the same between first and second editions, and the rulebooks are generally laid out the same, with the exception of these three changes:
Wise and Otherwise
Wises changed greatly between first and second edition. Instead of being skills in their own right, now they're just something you have or you don't, and they're not rolled directly to do anything.
Wises can assist in skill checks as normal; in addition, when a Wise is applicable and you're the active player, you can spend a Fate point to reroll a single failure or a Persona point to reroll all failures. When you have assisted on a success and a failure and spent both a Fate and Persona point on a Wise, you get a small boost; you can swap to a new Wise, or get a free check on a Beginner's Luck or skill related to the Wise.
Since they're not skills anymore, Wises have their own small section in the second edition rulebook.
All Together Now
Conflicts no longer split up the Guard between multiple allied teams. There's only two sides, one of which is usually the GM, though Players' Turn conflicts might only involve players. Some guardsmice can hop to the GM side if the conflict's more argument-styled and the Guard can play both sides of it, but much of the time they'll all be on one team. The rules do suggest that if circumstances arise where, for example, the patrol needs to both put out a fire and drive off the weasel arsonists, you can split the patrol and run two conflicts in parallel.
As a corollary, any guardsmouse can only receive two helping dice during a conflict, even if you're running a patrol of five where everyone could pitch in. (That's helping dice from relevant skills, not the bonus from wises or from other tools or supplies.)
The single "resolution" chapter in the first edition rulebook has been split up into the various elements of resolution: obstacles, conflict, conditions and recovery.
Wide, Not Deep
Recruitment has also changed a bit. Guardsmice begin recruitment with a fairly wide base of skills rather than being blank slates. However, all guardsmice make about the same number of choices during recruitment, instead of patrol leaders having huge amounts of checks to distribute. As a result, created characters aren't focused in a handful of skills but have a wide selection, able to perform well in a wide variety of conflicts.
The Wise rules by themselves are well worth using, but I've just summarized them for you. Character creation is also a little more streamlined for newer players.
If you need a boxed set, absolutely get the second edition one. I'll leave it up to you if having the other rules on reference is worth getting a second edition PDF.