The order reflects the judgment of the person speaking. If they think a woman is lazy but brilliant they're going to hire her, but if they think she's brilliant but lazy they're not. The term that comes after the "but" is always the more important criterion that leads to the final judgment. Similarly, if someone thinks a man is attractive but dishonest they're going to avoid being in a relationship with him, and if they think a dish of food is unhealthy but delicious they're going to eat it anyway.
There are other expressions that have basically the same function, and here we get to the topic of bridge tolls and the State Senate. For example, here is Senator Bill Perkins (79.8% of households car-free):
Bill Perkins of Manhattan has indicated support of Silver's plan but noted, "We have to proceed with caution, and maybe some refining."
Support but caution means that if something comes up he doesn't like, it's a no-go. Contrast this with Senator Duane (77.7% car-free):
State Sen. Thomas Duane supported the Ravitch report, though he was concerned that placing booths on the East River bridges might cause vehicles to back up into Brooklyn as drivers wait to pay a toll. Aside from that concern, he supported the plan and funding the MTA.
That "though" is like a "but," except it works backwards. The thing before the "though" is the conclusion. "Aside from" is like a "but" in that the last element is the deciding factor, but it comes before both elements. So Duane is saying tolls might cause backups but he supported the report. He has that concern but he supports funding the MTA.
These are important because they tell us where the speaker's priorities lie, and what conclusion they expect to reach. Of course the wise thing for politicians to do is acknowledge the concerns of all their constituents, but the "but" tells us how important those concerns are relative to each other. For Perkins, it's support but caution. For Duane, it's concerns but support.
So why is the guy representing the least car-dependent district in the State Senate, and probably the least car-dependent district of that size in the country, placing concerns ahead of support?