Why can a DMZ be regarded as a special perimeter, and why did web application firewalls become especially important for it?

Because it is itself a zone with a controlled boundary, every improvement developed over time for the classic perimeter (gateways, proxies, better firewalls) can also be used at the DMZ perimeter. Since the services offered on the internet are overwhelmingly web-based, the gateways specialised in web protocols — web application firewalls — turned out to be the most important of these improvements.

A perimeter is a system boundary whose crossings are controlled. A DMZ has two such boundaries, one to the internet and one to the LAN, so it is a perimeter in its own right, just one built for systems that must be reachable from outside. Everything learnt about strengthening perimeters therefore carries over: a gateway can sit at the DMZ boundary just as it sits at the company boundary.

Which gateway matters most follows from the traffic. The outer DMZ firewall opens HTTP(S) to the web server, and attacks such as SQL injection or cross-site scripting travel inside that allowed protocol, invisible to a firewall that only checks ports. A gateway that understands HTTP and inspects what the requests contain closes that gap.

Go deeper:

• Web application firewall — Wikipedia — history and function of HTTP-specialised firewalls.
• OWASP Top 10 — the web application risks that travel through an open HTTP(S) port.