The first difference I see is that the decision made with Timothy was from a position of strength - a choice made for the impact it would have on others - It was opening doors. The Galatians, on the other hand, were being pressured from the outside by so called Christ followers who made circumcision a matter of getting righteousness, not a way to build bridges to unbelievers. I think there are still a lot of customs that we have made a matter of righteousness today: rules about dress, worship, and use of anything that others have made an addiction out of...EXCEPT of course our favorites - fluffy foods, materialism, being "nice", and the three "C's" (cash, comfort and control).
This leads to the second difference I see: That nothing - whether circumcision, baptism, abstinence, or attending "church" on Sunday - can be allowed to define my righteousness. Paul says in 1Cor. 6:12 "All things are lawful for me, but not all things are profitable. All things are lawful for me, but I will not be mastered by anything." That includes religious things! The "circumcisers" were attempting to "master" the Galatians - to hijack their righteousness! Paul's statement to them says, "You can't be righteous because of Jesus AND following rules - only one can be Lord of our lives: Rules or Jesus. Take your pick!" (That's a really loose interpretation, but I thinks it's accurate)
Finally, I see so much of Paul's writing creating this tension, which is why Peter wrote of Paul in 2 Pet. 3:16 "...as also in all his letters, speaking in them of these things, in which are some things hard to understand, which the untaught and unstable distort, as they do also the rest of the Scriptures, to their own destruction." That's a pretty unique statement in Scripture, but it brings it down to earth for us. I think he is referring to people who look for statements in Scripture that seem to say what they want to say. Paul has plenty of ammo for them because he speaks into context so often - not in generalities.
How does this impact leadership? I think we need to check ourselves for ways we have reduced the message of the cross to rules in our own lives. We are called to apply the Gospel into the context of our life every day, but we have to hold fast to the grace of God, while holding loosely to the applications. May you find freedom to do this as you follow Jesus, and lead others with your life.