If operating properly, a pure sine UPS shouldn't damage any equipment and should even have 'cleaner' power than plugging your PC directly into the wall or into a cheap power strip. I'm curious in what way the UPS in the comments was faulty - if it was, for example, damaged by a power surge... odds are the computer would have been damaged either way. Also unclear if that failure happened while the UPS was on battery power, because otherwise they basically just act as a glorified power strip (and I assume surge protector but I haven't personally tested one to check). Plus, CPU+GPU being fried sounds more like damage caused by an internal PC power supply failing than what you would find in a UPS. In either case, many surge protectors offer warranty coverage for damaged equipment, so be sure to fill out that warranty card!
In terms of wattage, if your computer is using a true 600 Watts, than 650 Watts is a little close for comfort, and you should should consider getting a higher rated UPS. However, if your internal PSU is merely rated at 600 Watts, then your computer isn't likely to be using that much power (probably more like 300-400W, and 650 Watts should be fine in that case.
You generally want your power supplies rated to at a higher wattage than the actual amount of power you plan on using, as they will run cooler and more efficiently.