With evaporating or boiling water, the evaporation of the water takes away an enormous amount of heat. It takes one small calorie (which is 1/1000th of a food calorie) to raise the temperature of one gram of water one degree C, but it then takes 575 calories to boil that water into vapor. In other words, if you add heat to almost-freezing water until it starts boiling, it will take almost six times as much heat energy again to completely boil the water away.
With your lid off, it becomes easier for the water to evaporate away, which extracts a large amount of heat energy from the water, keeping your example pot at a simmer. Put the lid on, and you make it harder for the vapor to escape, so less heat is removed, so your pot heats up further to a rolling boil.
But, since you're adding the same amount of heat to both pots, in both cases the pot will heat up until the heat lost per time equals the heat added. Although the majority of the heat loss in both cases will be the heat of vaporizing water, having a lid lets you have proportionally more heat loss through conduction to the surrounding air (which is why, with a glass lid, you'll see water condensing on the underside; that transfers heat to the lid, which then transfers heat to the air). So, you'll lose somewhat less water with a lid, but I'll guess there wouldn't be a large difference.
In your case, my guess is that you'll get more reliable boiling with a lid, especially a glass one where you can see the contents. And, the somewhat reduced water loss will be a bonus.