Rented stuff belongs to someone else. I need to take due care, and if I do happen to break it, I usually have to pay more than what getting the equivalent thing off craigslist would cost me.
Also, when you rent, you usually commit to renting for a certain number of months (or sometimes years) - in many ways a lease is much less flexible than owning, because I can't sublet, and I can't force the landlord to lower the price, so if I want to leave, and the market prices of the rental is lower than what I paid for it, I'm potentially on the hook to pay off not only the difference between market value then and market value now, but instead the entire value of the lease.
Another example; a while back I stepped on my thinkpad. Now, being a 200lb guy, the monitor cracked.
Will insurance cover "A fat man stepped on it?" If they do, they'd probably pay retail to replace it (lenovo wanted $700 to fix it, and that'd be factored into premiums.) I think I ended up paying $80 for a new LCD and spent two hours installing it. Probably less hassle than dealing with insurance would have been.
Insurance is best for risks that are beyond your capability to deal with... Usually for small things, it's cheaper, long term, to deal with it yourself. And unless you /enjoy/ bureaucracy, it's usually less work, too.