The difference in the “compromise” is a matter of semantics. The first mandate directly required the religiously affiliated groups to buy health insurance that provides contraceptives, sterilizations, and the plan B pill. This mandate was blatantly against the First Amendment and is still actually the one that has been written into the bill as of the last time I checked even though the White House has said they have a compromise. They haven’t changed the bill to the compromise yet.
The compromise basically said hmm alright we won’t tell you directly to buy health insurance that provides all these things that go against your belief, we will just make it so you have no other option but to buy health insurance that does provide it. They have basically decided to mandate that all health insurance providers provide these services.
If all insurance companies are required to provide said services then it only follows that everybody (including religiously affiliated organizations) is required to pay for said services, and thus by getting rid of all other options requiring everyone to pay for them.
Constitutionally this mandate falls short because they cannot show that said mandate is necessary or even the best way to make sure everyone who wants these services can get them or that right now everyone who wants them isn’t getting them for little to no cost. In other words there is no express need for this mandate or any compelling state interest to do so and the First Amendment should not be trampled on for something so frivolous.
The second way in which it fails is that the new requirement does not apply to everyone. Thousands upon thousands of waivers have been given out mostly to Obama supporting groups, and it thus follows that the law is not being applied equally to all and the Church can claim it is being unfairly targeted.
In short basically instead of requiring them directly to provide a service, the White House made it so the only option they had was to buy the service they wanted them to buy.