Yes, good LEDs put out more light.
That light eventually also turns into heat..
Photosynthesis while endothermic is very inefficient so negligible.
Energy that's stored inside the plant doesn't turn into heat (only a very tiny amount).
So 1200W of LED practically produces just as much heat as 1200W of HPS.
But the radiation of the 2 sources is different.
HPS has some IR which will radiate right on your plants and you can assume all of that will turn into heat the moment it hits your plant.
So even if room temp is equal, plants under HPS will have warmer leaves because of the IR radiating on the leaves. Something to consider.
Even funnier if you're trying to grow with incandescent, should be very hard since incandescent bulbs are extremely efficient IR producers and will easily fry your plants even if ambient temperature isn't very high.