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.