You can see for yourself the changes to the "Hub and Spokes" rules, as they are the first 90 pages, and are included in the preview at http://www.burningwheel.org/?p=276. Under the word "Here" in the fourth paragraph on that page is the download link for that preview.
Luke has mentioned that there are numerous small changes to Fight, specifically the processes for range setting, Duel of Wits, and Range and Cover.
Sorcery no longer adds will dice to rolls to cast, reducing the difficulty of advancing Sorcery as one no longer needs insane Obs to get Difficult and Challenging expenses.
Perception is no longer open ended.
Let it Ride drops the bit about referees cheating.
A significant change in terminology: "Graduated Test" replaces the older "Open Test", avoiding the frequent confusion between open ended rolls and open tests.
Some minor changes to the difficulty/test type table at the low end.
Many lifepaths have been revised.
I've not yet gotten BWG yet, only the preview, so I can't be much more detailed, but I'll note that the hub and spokes text is not readily different at first glance, but some small changes are present.
References
http://www.burningwheel.org/forum/showthread.php?11076-Burning-Wheel-Gold-Errors-and-Changes
I will be quoting from the Burning Wheel Gold rulebook.
Focus
Fight for what your character believes. Everything else in the rules tells either how to craft that character's beliefs or how to fight for them. (Foreword by Jake Norwood)
At its core BW is a character oriented game. It is narrative first and rules second but the narrative revolves around the characters and their relationships. If the characters are just there to experience the story without being its source, that's not doing BW as intended.
Player skill
Burning Wheel's core mechanics, advancement and Artha rules demand more-than-usual attention from the player. Skill or stat advancement isn't an afterthought, but rather a crucial part of the game. The decision to solve a problem with cold steel or silken words isn't just one of better numerical values - it's a question of who you, the player, want your character to become. (Foreword by Jake Norwood)
Burning Wheel is demanding for players. This doesn't necessarily mean BW is for experienced players, it's for invested players. They are expected to know what their characters are capable of and what they want to accomplish. Experienced P&P players don't necessarily do better because BW does some things significantly differently than other games. A fresh player might cope better than an experienced D&D veteran who is accustomed to crunching numbers and skimming supplement books and playing tactical combat.
The GM may veto a character that he feels has been min-maxed. Simply chant these magic words: "I don't think he really fits in with the game or the other characters." (p. 106)
Rules
It is the philosophy of the system that some tasks are either too complex to accomplish without help or simply too difficult to accomplish at all. Sometimes, high obstacles will force players to be creative and think of another way around. (p. 15)
The game's basic rules cover combat and non-combat situations in equally detailed manner. The main part of the basic rules is the character advancement. The learning-by-doing of skills, the difficulties, the helping rules and the Artha system are tied together in a very tight knot at the games center.
Beliefs, Instincts and traits (BITs) are the primary conduit between the player, his character and the artha system. Setting out BITs for his character, a player states to the GM and the group what his goals in play are for this character. (p. 61)
All expert rules are there just to have more detailed resolution for certain situations. There are advanced rules for verbal disputes (Duel of Wits) tactical move and shoot archer battle and small group tactics system (Range and Cover) and an extremely detailed melee duel and brawl (Fight). They are very detailed and not meant to be used for every fight or argument.
Vincent's advice ["Roll dice or say 'yes'" by Vincent Baker of Dogs in the Vineyard] is perfect for Burning Wheel. Unless there is something at stake in the story you have created, don't bother with the dice. (p. 72)
Other rules are also only meant to be used only when the beliefs of the characters are at stake. Money and wealth is abstracted in a separate attribute. Travel and other stuff is glossed over unless it's important to the beliefs. Low stakes fights get resolved with a simple ability test.
Pace
The rules go equally well with slow political plots and fast action scenarios. However the focus on advancement as the main game engine demands the attention to stay with the same characters for some time to get the game really going.
Best Answer
Let's start with the similarities:
They use the same dice mechanic.
They use script 3 actions in the conflict mechanics.
They both use Persona and Fate points.
Resources and Circles work very similarly, but not identically.
Both have the end of session workhorse and MVP votes.
And the differences
Mouse Guard has...:
Structured play. GM turn then player turn. GM Turn is 2-4 (sometimes more, but "normally" 2-4) encounters on rails. Player Turn, each player in turn narrates until the GM thinks the action narrated absolutely requires a test. Test is made, and another player narrates from there. You have to spend checks to make tests; if you have no checks left, you can't make a test, and so don't get to narrate.
Further, the campaign is structured into missions, seasons, and years. With specific mechanical effects.
Conflict, the "combat" system, is used for any extended conflict, not just duels of wits, nor martial conflicts. I have used it for a few off-the-wall items, including moving a bee hive, building a dam, and crossing a burning prairie. It's only got 4 action types: Attack, Defend, Feint, Maneuver. Further, the GM can pick which skill fit which slots when a non-standard type of conflict is desired. (Such as Bee Wise and Beekeeper for moving the hive...)
Artha is renamed Rewards. No Deeds Artha. More rigid awarding schedule in MG.
Conditions: tired, hungry, angry, sick, and injured. All are boolean. No damage done during conflict; inflicting conditions has to be part of the intent to inflict them. No multiple stages of injury.
"Checks" - nerf yourself using a trait, earn a check. Break a tie against yourself with a trait, get two. Each is one action in the player phase.
Traits - are unlike BW traits. They have a level (1-3). All of them can be used to your benefit but are also disadvantages as well. Only allowed to have 5 total traits.
BIGs not BITs: Belief, Instinct, Goal. All are functionally BW Beliefs; MG instincts are not "Rulebenders" and are mechanically more like BW Beliefs. There is no equivalent of BW instincts, and traits (as mentioned already) are not mechanically the same.
Skills - far fewer, and of the 30-some in the book, plus the player defined wises, no character can ever have more than 24. Wises are far more common than in BW, and more explicitly can create situations/narrative truths. The skills themselves are also generally much broader than in BW. There are only two combat skills - Hunter and Fighter.
Attributes: only two - Health and Will. They are used differently than in BW, as well.
Advancement - successes and failures, rather than Routine/Difficult/Challenging. Skills limited to level 6. Resources and Circles to 10. Nature to 8.
Nature - is related to the "emotional attributes" of Faith, Greed, Hatred, etc... but everyone and everything has a nature score. It can be invoked with a persona to add dice, as well. Nature has 2-4 descriptors, by species, as well, that affect how it may be used.
Stock Choice - None: all PCs are mice. So all PCs have the same Nature descriptors... most critters don't have Mouse Nature, but still have a Nature score and use it for everything.
Magic - None: no magic system. No magic items, either.
Trait Votes - Trait votes, however, are regulated by the seasonal mechanics - they are rare, and much more limited.
Weapons - don't do damage, but provide conditional bonuses to conflict skill(s). Not all weapons are for combat, either. A juicy rumor is a Duel of Wits weapon.
Teamwork rules more detailed than just BW's Help dice - specific effects on Conflict, as well - and much more important since...
I am Wise - FoRKing is limited to one wise per test only. No FoRKing of non-wise skills. Use linked tests instead.
Structured encounter design - some kinds of encounters have specific mechanical rules (tho' they're very straightforward and not many) - these have preset difficulties by season. Other encounters can easily be balanced against parties because of the nature and help systems.
Scaling Rules seriously limit what you can hurt or kill.
Simplifications in Mouse Guard
Only two kinds of Artha.
Limiting FoRKing to one wise-skill only.
4 actions in one conflict system used for everything other than rolling single tasks.
No different levels of injury
60-some skills instead of 200+
Commentary
Mouse Guard is close enough that your skill at belief writing crosses over well, and your understanding of the meaning of a given skill level is close enough to cross over either direction.
The play experience, however, is VERY different. So much so that BW and Mouse Guard don't feel much alike at all, and most players of both have a clear preference for one or the other.
The use of a single conflict mechanic is very different as well - there is less detail to it than BW's Fight or DoW, so it's faster and plays smoother with fewer lookups, and it replaces both of those.
Mouse Guard, as Luke Crane mentioned in an interview this month (December, 2011), started off with BW, then evolved away into its own thing in the playtesting and writing process.
Also, I wrote the comparison page on the BW Wiki... Comparing in point form Burning Wheel Revised, Burning Empires, and Mouse Guard, as well as Torchbearer another similar game.