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.