Indeed, you can delete the wrong quotes and substitute with the right ones. But Word special characters are not limited to quotes. For example, in Word, if you type:
This - is a test.
That "dash" will be changed into an "em dash", which looks the same, but is longer. If you didn't know to change it after you copy/paste into the web form, it could be a become a funky character.
For the record, I did use Word (they copy/paste into Notepad and fix) for my AMCAS and my secondaries. I think it was fine, for the most part. The only time I think there was a problem was with NYU's application. They insist you type your essays in their web form and not copy/paste from Word, but they had quiet a few essays. I didn't listen, thinking I knew what I was talking about and knew about the special characters that needed to be changed. Everything looked good and I submitted.
After submission, it presented me with a copy of what I submitted. My essay was clumped into one giant block of text without paragraphs (I don't think this is the result from different interpretations of Carriage Return/Line Feed. Yes, even hitting Enter on your keyboard to create a new line is represented differently in different systems:
http://en.wikipedia.org/wiki/Newline). I don't know if typing directly into the forms would have changed the result. I don't recall there being a way to make changes after submission, so I just lived with it. I never did get an interview from them, but I doubt that was the cause. I hope not, at least