It's not about saving money. As this thread shows, charity is akin to a good PR move. Paying taxes is seen as neutral, at best. If you're "spending" either way, you might as well get something for it.
It's better than spending the money on yachts or luxury goods, which is the other way people signal status. I'm a big fan of conspicuous philanthropy over conspicuous consumption.
Debatable considering that donating art works (whose valuations can be, let's say, subjective) for tax purposes is a fairly well known way to reduce your tax burden.
It's still money gone from your pocket. It doesn't save you money overall.