It matters because, as an atheist who abhors that religious organizations are tax-exempt
I think there are good tax-policy reasons why religious organizations are tax-exempt independent of their status as religions. There are many, many, tax exempt organizations out there with often contradictory policy objectives. I think if you make the policy decision that one non-profit org is tax exempt, you need to apply that status to all of them. The income tax is fundamentally a tax on profits- a group that has none wouldn't pay much tax anyways. You would also into a ton of problems RE business expense deductions if you tried to tax religious orgs. For example, IRC Section 162 sets forth a deduction for "ordinary and necessary" business expenses. You would end up with constitutionally dubious cases where the government had to decide if buying things like holy water are "ordinary and necessary" to the functioning of the church. Therefore, those who support a separation between church and state should support tax-exempt status for religions.
Note that there is a difference between merely tax-exempt organizations and tax-exempt organizations that you get a tax-deduction for donating to. They actually fall under different provisions of the tax code. I think there are better arguments for why religious groups should not receive that status.
Well first let's be clear that we're talking about 'charitable not-for-profit' organizations, and not 'non-profit'. As one IRS agent once told me, 80% of all businesses are non-profit.
I have no issue with charitable not-for-profit organizations who perform service for the public good, except a lot of religious organizations aren't exactly charitable not-for-profit engaged in services for the public good. The Roman Catholic and Mormon churches are the two examples that pop first into my mind, both with enormous coffers of wealth. The money that these organizations receive isn't being returned 100% (minus administrative expenses) to causes for the public good such as finding a cure for cancer or clothing the poor. The only reason these organizations aren't being taxed like they should, is because of their 'religious' foundations.
I firmly believe in a separation of church and state, and for that very reason state shouldn't give church a tax exemption. It's bad policy since it encourages religious beliefs (which is nothing more than a poor man's old-school social service) over substantiated, researched, and modern science. Religious organizations are welcome to use those funds they receive for the public good and not be taxed on such, just as a psychiatrist is allowed to write-off funds they give to qualified charitable organizations for the public good (such as the National Autism Association). Religious service however should be considered just like any other service and taxed just as a psychiatrist has to pay taxes on the profits they make and property taxes on buildings they own to provide those services.
And yes, I'll be the first to admit that my ultimate goal in these policies is to see that the silly institution of religion disappears in this country because religion has proven time and again to do nothing but harm and hold humanity back from progress.