what are the main differences between WordPress installed on a LAMP server, and a WAMP server.
From the standopint of running WordPress, there are none, except that Apache may be less stable on Windows than on Linux.
I'm trying to get a customer to buy a linux server, but they want to use Windows.
This is potentially a problem. If they want a Windows server and they actually have reasons for it, it is possible they have certain applications they want to run on it, some of which may require IIS, which means that you won't be able to run Apache on it (unless you want to deploy Apache on a non-standard port)...
Although to be honest with you, an in-house Linux server suitable for running WordPress could actually be a decommissioned PC (as little as 256MB RAM and 20 GB hard drive is enough for a functional LAMP server), so there is no need to buy a server to run WordPress; you can pick one up in a computer junkyard.
What changes for WP, from one environment to another?
For WP proper, nothing. There is one important difference PHP-wise though. On Linux, PHP integrates with a mail transfer agent (MTA) such as Sendmail and queues ougoing mail for processing by MTA directly (much like
pine and other old-time mail clients used to do); on Windows, PHP has to send mail via SMTP, which is slower than MTA queuing.