Power dissipation is directly related to the size. For a given size, and the same power being dissipated, the temperature rise will be the same for a 1W rating and a 2W rating. However, a 2W part dissipating 2 Watts will be hotter than if it was only dissipating 1Watt. A 2W resistor is capable of the higher temperature from dissipating the higher power and still remaining within the rated resistance tolerance.

I quickly checked a Venkel catalog (because it was within arms reach) and they have both 1W and 2W resistors in the 2512 size. In fact, they have two wattage ratings on most SMT resistors.