Because this is a spending bill the house has the last vote because the house "holds the purse strings". The House will vote first once passed it will move to the Senate if any changes are made by the Senate and amendments added it will need to go to conference committee this is a committee of House and Senate who work out the differences. Then the house will need to pass it again before it goes to the president
So, Snow doesn't like some of the things that Peloser put in the bill. So, she votes to remove these items and it happens. Then it could go back to the committee and the committee could add it back in by rectifying the difference. The House votes again, it passes and then goes to the President. If that is right, I don't like the way that is set up.
I thought both houses had to agree, but I may be wrong. Everything else is correct. These joint committees are only as strong as the members who participate and can be quite contentious.