Apache is just a web server program - when you start it up, it just listens on port 80 (for HTTP) and/or port 443 (for HTTPS) by default - or whatever other ports you specify in the configuration file.
Yes, Apache will work with PHP - but you generally need to install PHP separately in addition to Apache - but making sure that the version of PHP that you install is compatible with the version of Apache.
MySQL is a database, so you will need to install MySQL separately from Apache - it should work just fine with PHP though (as long as the version of MySQL that you use is compatible with the version of PHP that you already installed).
Yes, Apache will work with AJAX - provided that whatever you write your web applications in (e.g. PHP) supports AJAX - and yes, Apache works with XML too.
In terms of SOFTWARE, there is probably minimal difference between a web host on the Internet and an Apache server running on your PC (they will work in exactly the same way). But for practical purposes, a web host will probably be running a fairly high spec hardware configuration in a data centre connected to a very fast Internet connection, with things like UPS backup if the power fails - and generators on standby to kick in if necessary - whereas if you want to HOST an externally accessible website from an Apache server running at home, then you'll either need a static IP address from your internet service provider, or will have to make use of a Dynamic DNS service - and you'll need to make sure that your internet service provider doesn't block port 80 too - and you'll also need to configure port forwarding on your ADSL or cable router so that traffic coming in to port 80 on your ADSL/cable router gets forwarded to your computer running Apache - then there are security considerations too, such as making sure that you run a suitable firewall and only open ports to the outside world that you need to.
But you'll probably be really disappointed at the speed with which any websites you host load when you try and access them from someone else's computer (not in your house) if you're hosting them from the end of an ADSL or cable connection at home though - as the upstream speed is normally limited to less than 1Mbps - if that - compared with a web host where the connection speed would normally be at least 100 times as fast as that.
So although Apache installed on your PC at home will probably work in exactly the same way as Apache at a hosting company would, you'll probably find that hosting at home gives you vastly inferior performance compared with using a hosting company - simply because of the upload speed to the Internet being a hell of a lot slower via an ADSL or cable connection than it would be at a web host's data centre...