Functionally, the only difference is they can't pay dividends to shareholders. Non-profit's can (and do!) turn profits into executives bonus's or perks, or bloated organizations, etc.
Why do you feel a non-profit would run a service better?
In the UK, it's easy to set up a non-profit as a Company Limited by Guarantee, which has no shares. Such companies can still make profits, but they can't necessarily be distributed to Directors. It doesn't prevent them from making profits or spending them any other way.
For a comparison, see past discussions about Post Haven (re: pledges to "never be acquired") . Had they set up as a non-profit from the beginning, the purpose would have been clearer and enshrined in the company's legal structure. Presumably fewer people would have had cause to question the founders' motives.
Of course, non-profits are not created equal. There's IKEA and then there's Mozilla or EFF.
The IKEA example is interesting as I think the aim of the convoluted legal structures (inc a non-profit somewhere) was to allow one person to have almost total control over the company.
Edit: For anyone interested in the IKEA web of companies, you read the following article: http://www.economist.com/node/6919139?story_id=6919139