All other things being equal, yes — mostly. In terms of exposure and depth of field, they would be as close to functionally equivalent as things get in the real world.

The shape of the aperture blades will have some impact, so in some cases you could probably tell by looking closely, even if that were the only difference. But in the real world, those lenses probably differ in other more significant ways.

For example, it is usually the case that constant aperture zoom are higher-end, so you might have reasons to buy one other than this particular feature. It's likely to have better environmental sealing, and will probably better correct for various distortions and artifacts.

In fact, in the real world, I expect that even though stopping down a bit usually improves technical image quality, the constant-max-aperture zoom wide open at the wide end will beat the variable lens one stop down. On the other hand, the variable aperture zoom will likely be lighter and cheaper, which can both be important considerations.

Also, as Itai points out, cameras use the widest aperture for composition and focus, so the viewfinder may be a bit brighter and autofocus a bit faster with the lens with the wider aperture (so possible advantage to the variable max aperture lens in this case).

You probably will find How do constant aperture zoom lenses work? to answer a lot of your question,as well.