By layout and organization, MYSQL is an RDBMS, but not at the level one would want in a large scale enterprise system - yet. Last time I looked (last winter), by my notes it was missing views, stored procedures (?), userdefined functions, some types of joins, and full transactional support.
Most web apps, short of large scale enterprise apps, won't care about any of that except perhaps the stored procs and views. For the typical "flat" collection of tables found in may web apps, MYSQL is faster than MSSQL. MSSQL is more reliable (for db related faults), but includes more overhead to get that reliability. Bad sql and bad db handling in the code will hut performance much more than your choice of an RDBMS.
There's a myth that MYSQL is free. It is, but only for personal use or for non profit orgs. Any other use, including bundling it in an app you sell - e.g. desktop/lan/private web server/reseller web server - or offering it as part of a web hosting package requires payment of a $400 license fee.
I agree with the observation about PostGreSQL being better than MYSQL, but some of the commericial flavors of MYSQL are prety good also.
If I were to deploy a commerical app that expected under 5K db hits per day, I'd look at MSDE. Right now, it's not the snap to deploy that Access is, but MS has committed to move it in that direction. I'd expect to see Access go away in the near future.
One analogy I read about MSDE and it's throttled version of MSSQL was that it could run all week servicing a Human Resources DB for a company with 1000 people. But, when they all hit it just before 5 on friday to do their time sheets, it'll go down the tubes.
Edited by dpyers