In Dallas, you'd be called upper middle class. In SF, people lecture you that you're "rich".
Post-tax, the difference is only ~$45k. Your rent is going to be an extra ~$20k. Your cost of living is going to be an extra ~$10k.
Okay, so you've got an extra ~$15k, maybe 20% more purchasing power. But this only works if you're single.
If you have kids or a dependent, you're going to lose all of that to rent and expenses (and probably more).
And, until this year, it's not like you could just take your $180k job to somewhere cheaper. If you got a job somewhere cheaper, you'd get paid less to where it wouldn't make much of a difference financially.
Really - the only benefit is how regressive the 401k system is. You can dodge taxes on your high marginal tax rate in SF. So you're much more likely to contribute to your 401k - which will almost certainly work out for you in the long run.
In the short run, you're likely to pay for it significantly with a reduction to your quality of life.
Rich is when you don't need to think about any of this shit at all. People keep trying to put a dollar amount on what being rich is. We're all middle class arguing about how much we make PER YEAR. The truth is you're only rich if you control enough wealth that you can not work for a decade or more.
For example my step father start a trucking business that required him to work from 5 am to 5 pm every day for more than 10 years. Slowly he added more and more drivers. Recently he sold his business for 1.1 million.
Now is he rich? I personally don't think so. They live in a modest townhome in a low cost area. He just has a low burn rate relative to the amount of assets he has. There are way larger houses all around them.
And the truth is most normal people will agree with me because that's what their lives are. What you make per year can't be used as a metric for wealth. It's just earning potential not actual wealth.