The two-interface (aka two-tiered firewall) architecture is all about defense in depth for your internal network. The dirty and clean scenario makes traffic go through 2 different sets of firewalls. The internet-facing firewalls protect your DMZ against external threats, while your internal firewalls protect your internal network from potentially compromised DMZ systems.
The thought behind having two sets of firewalls is that if the Internet-facing firewalls become compromised, or if somebody accidentally opens up too much access, the inner set of firewalls will still protect your core network. Quite often the 2 sets of firewalls will be from different manufacturers so that no single exploit will compromise both sets of firewalls. With one set of firewalls you are putting all your eggs in one basket.
You don't have to have to interfaces, but there are advantages:
- Measurements: keeping the dirty and clean traffic separate means you can measure the quantity of the two types of traffic
- Network simplicity: It's somewhat easier to keep routing straight with two interfaces.
- Minimize the mixing of dirty and clean traffic: The goal is to reduce the traffic that directly transits between the external and internal firewalls. The goal is to force all traffic through a device in order to transit between the tiers. Ideally no traffic will travel directly between the internal and external firewalls. Web traffic will go through a proxy for example. The through it that the more devices that traffic has to travel through the more chances to stop malicious traffic
The downside to it is increased complexity and management overhead. It also rarely makes a difference to a company's IT security, so it often seems like it's a lot of work for no benefit.
However, it makes a difference often enough it's generally worth doing. It also makes auditors happy; you'd need a lot of justification to show that having a single tier is sufficient for your organization, especially since somebody thought that your organization needs two sets of firewalls to begin with.