I was recently asked to explain the difference between the two and I thought I’d share it here so I won’t have to retype it another time (this wasn’t the first time I’ve typed it out and not saved it
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Okay . . . the two scenarios you want to avoid in a relationship are You win/I lose and I win/You lose. When someone digs in and demands their way you might occasionally give in and not make a scene, but you can’t build a strong foundation on this. Traditional female-only submission is supposed to *always* fall into “he wins/she loses” but she’s not supposed to feel like she’s losing because when he wins, she wins
And then the woman’s spiritual maturity is judged by how well she embraces “she loses” as “she wins”.
The “better” solution, and what many marriages strive for, is compromise. Both win BUT both also lose. You give a little to get a little. You both come to the negotiation table in the hopes that you will get your most important things and you are willing to give up the least important. So everyone leaves “accepting” the answer. This is often mistaken for “mutual submission” based on the idea that “submitting” means “losing” and in this case you both lose 
BUT mutual submission is actually rooted in the idea of resolution. In resolution there is NO agenda anyone comes to the table with EXCEPT the best for everyone–and, ideally, what God wants (Though this can be done without being Christians if you both want what is best for everyone involved). Once you come together to find a solution you *together* define the problem. What became an issue as “how do we get the baby out of our bed” may be redefined as “how do we all sleep best?” You don’t just have one person redefine things–you sit down and talk about the issue until you see it from the angle of everyone involved. It might start with “I would love to go to San Diego on vacation, and someone responding, I was thinking Mississippi so you realize
we have a problem–let’s solve it. And you REdefine the problem as “Where can we go on vacation that we can do X and Y” as the two things you realize you’re expecting from both places. Then you may find that one meets both hopes or you may pick an entirely different place 
So once you redefine the problem to take EVERYONE’s needs into account, you brainstorm. NOTHING is off the table in brainstorming and you don’t evaluate at this time. The reason is that silly ideas often poke someone’s creativity and you can end up with some better ideas than what you thought was *the* answer before you brainstormed. The idea is to think outside the box and get creative and crazy.
THEN you evaluate the brainstorming session. Cross off the ludicrous; cross off the crazy, dangerous, would never work. Then really look at what you have left. Narrow it down not to what he wants or she wants–but what meets the needs of solving the problem. “Would everyone sleep well?” then keep it on the list.
THEN you evaluate the possible solutions–everything you have to choose from will meet everyone’s needs. And through the process you’ve learned about other concerns and desires that each person has. Maybe the real concern is that he wants to snuggle his wife and so cosleeping isn’t a concern if there’s a way to move the bed or add something that will allow baby to not sleep between them, just for example.
What *I* add to this process is specific criteria that *I* believe to be Biblical.
1) Does the Bible specifically address this issue? If so, regardless of personal feelings you do that.
2) Are there Biblical principles at play? If so, how does that play into your solution?
3) If neither of the above is yes, then you are free to do what you want and be okay 
The reality is, where there is disunity, there is chaos. God is not in chaos until we surrender it to him to bring order to it. THAT is the point of mutual submission–it’s the PURPOSE of mutual submission. When you are in disagreement it makes little sense to go with one person *or the other*’s ideas. That’s just deciding which human plan you’re going to go along with. Why not seek the greater good? The God plan! THAT should be what you’re looking for. So start this process in prayer asking God to stand in the midst of your chaos and bring order 
IME the resolution process exposes a few things.
1) the condition of the hearts involved and how submitted to the Lord each person is
2) the real problem which quite often has an obvious answer once you identify the actual problem
and
3) how united you both really are once you dig through the muck that becomes the focus when you are trying to compromise or win.

I felt so bad for him. And what’s her name from Mercy West–that was nice to see.