Access is for light weight stuff. One could argue MySQL is for light weight stuff too.
Access, the last time I checked, supports a really row number of concurrent users. Meaning if a bunch of people hit your website at one time, Access will choke and take your site down. MySQL will choke at 100 (I think - there may be a way to tweak this as necessary).
Access is pretty slow in terms of query performance. MySQL on the other hand is pretty quick.
Access requires a license from Microsoft to use. MySQL is open source and freely available.
Yes, Access is pretty and easy to use. With the right tools, MySQL (and other databases) are also pretty easy to use. Although they still require knowledge of writing SQL by hand. I could also mention Access's funky query language, but that's more personal opinion then anything...