A voltage stabilizer is a device that stabilizes voltage. More often, it is called a line filter. It prevents transients (high voltage spikes, etc.) from making it through to the delicate equipment connected to it.
A UPS (Uninterruptible Power Supply), on the other hand, provides power (for a short period of time - often 15 minutes) when the main power source goes down. This gives you a chance to do a proper shutdown, such as saving files and stopping the computer the correct way. (You should never "just turn off" a computer.)
While a stabilizer/filter can actually persist the power, that persistence of usually only a half line cycle - 8 milliseconds - and that is not enough time to do a proper shutdown.
A UPS usually also functions as a stabilizer/filter, protecting the equipment.