Logic tells me that both should be the same because they both just strain the grounds. Is there any added effect of pressing down the plunger in French press?
I would say no, there is no macroscopic difference.
Actually, some people (notably James Hoffmann) advise not to press the plunger all the way down since it agitates the grounds and this could result in having more fines in the cup. You are perfectly fine using a strainer, but depending on how big the holes are you could get more fines.
The only difference could arise by squeezing the grounds with the plunger. Maybe that way you could get an additional tiny fraction of extraction, but I believe it would be more or less undetectable.
In any case, it's a simple experiment you could try at home to convince yourself!
In terms of extraction (as in "Total Dissolved Solids") there is very little difference. However, in terms of "mouthfeel" there is a very noticeable difference, which I'm sure everybody can taste when presented two cups side by side, feel free to try for yourself if you have friend with a second french press. Pressing down the plunger undoubtedly agitates the coffee, even if you do not press through the grounds themselves. This results in more suspended fines, which yields a cup that feels sandy and gritty. If you use the plunger only as a strainer and pour gently, the resulting brew will feel clearer and crisper, since more of the suspended material will have fallen out of solution.