I want to save a logo as PNG-24 with a transparent background. Is it completely the same outcome if I
Save for Web in Photoshop and
Save for Web in Illustrator?
Yes and no, once the image has been turned to pixels yes. However there are more than one way to turn the vector drawing into rasters, which is where you can get difference by purposefully doing something differently.
Why you would want this is a bit over scope of the question. Lets just say that i am not a big fan of the antialiasation most mainstream vector engines produce so i might choose to use my own filtering of data samples.
I must say that "Save for web" is misleading.
It does not "save" a file, it "exports" it.
Saving is preserving the file, the format, the size, the resolution, the layers.
An export is a transformation of the file, not only in file format but on other aspects. It will flatten any layer, will define a final pixel size, sometimes resolution, some tines will change the color mode. On a vector file it will rasterize all the elements of the file.
Is it completely the same outcome if I Save for Web in Photoshop and Save for Web in Illustrator
The outcome, probably; depends on what you had in the original file. I will assume that on Ilustrator you do not have vectors, but some raster images. There could be some little differences, as mentioned by joojaa, the anti-aliasing, and probably the resolution if some elements are escalated on illustrator.