For simplicity sake I'm going to say that I'm making a hit counter.

However, because I have Google Ads on my page, the 66.249.73.XX IP of the Googlebot pops up on my logs a few seconds after a genuine hit because the ads have appeared. Obviously, the IPs of all other web crawlers that visit my site are logged as well.

Is there any way (through PHP preferably or I'm open to other methods) to differentiate between a genuine user visiting a site and a web crawler's hit? I want to only add hits for genuine people.

Blocking crawlers through robots.txt or any other method isn't wanted.

You can distinguish between bots and normal users from posted http headers.
normal users would have words like firefox,ie,opera,Gecko, etc..

user-agent Mozilla/5.0 (Windows; U; Windows NT 5.1; ja; rv:1.9.0.8) Gecko/2009032609 Firefox/3.0.8 

search bots should have some thing different mostly including the word bot

user-agent Mozilla/5.0 (compatible; Googlebot/2.1; +http://www.google.com/bot.html)

You can't get 100% accurate data but you can over time improve your statistics accuracy by keeping adding search bots headers to your script to exclude them