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.
I took a shot at it purely as a design experiment -- to see if I could strip Burning Wheel down to its core and keep it intact. We ended up with a rather different, unique game in its own right. I like it very much.
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.
Best Answer
There are a few conceptual things to remember here:
The full quote from that page is
These are very loose guidelines, not hard rules for all situations. Further, they're guidelines for setting Obstacles when nothing else gives you the number. When you know that something "should" be "kinda hard" or "very hard", this part of the book gives you an Ob that goes with that gut feeling. This is not trying to tell you what the odds for actual rolls will be — that will depend on how many dice the player is actually putting into the Test and is fundamentally not the GM's business to worry about, but the players'.
Nature is not the Obstacle, it's a rating that gives you dice to roll. The snake's Nature 7 means 7D, not Ob 7, and 7D in an opposed roll will average 3.5 successes. That makes fighting the snake hard, but not impossible.
Guard mice work as a team, helping each other. They also have gear that give them bonus dice, wises that make them wise and give them bonus dice, and circumstantial bonus dice. They can also tap Nature to get even more bonus dice. The Obstacles are high, but a mouse has a lot of options if they need to succeed.
They just can't use every advantage on every roll, so they have to think hard about when to ensure success and when to endure a setback. Which brings me to…
Even guardmice fail. In fact, in Mouse Guard they must fail in order to advance. They can't ever fail if the Obstacles aren't high to begin with. Choosing your failures is a key part of how strategy works in Mouse Guard.
The example of Liem fighting the snake on pages 109–10 (original printing) illustrates many of these. Teamwork matters, your gear matters, and choosing when to use your scarce resources can turn a very hard Obstacle into an easy success, if you choose well.
A key conceit of the dice system in the Burning Wheel family of games is that you can't win interesting challenges reliably, if you don't care enough to fight for it. And there's no point in rolling if the outcome isn't uncertain and failure a real possibility. Trying is modelled by using up scarce resources when you want to "push" for success; and a real possibility of failure is ensured by never giving weaker Obstacles than the situation realistically would call for.
There's no point in striving and fighting for what you believe in if it's easy, so Mouse Guard doesn't make it easy. It makes you choose. As GM it's your job to set up the Obstacles so that they're real challenges, not guaranteed wins. It's the players' jobs to step up and either rise to the challenge, or fall short and suffer a setback.