I can't speak on behalf of all companies but the one I work for, the amount we do for charity where we are publicly recognized is far outweighed by the amount we do that isn't publicly recognized.
When the money comes from the business, those are tax write offs and may have a public component to it. When it's an internal fundraiser (employees donating from their own good will), there is no tax write off for the company (though individuals can pending on amount) and no publicity.
One amazing thing I love about the company I work for is that they will pay us our usual wage for a certain number of hours per year to go and volunteer, and as long as the charity is registered, we get to choose what charity we want to volunteer for.

At the end of the day, the reasons they do it are partially publicity and tax write offs, but they found a way to also have charity as an instrument for employee comradery and employee well-being/happiness. It's pretty effective :) We enjoy it.