Perimeter router, internal router and firewall

Perimeter router is a type of router which is used for the outside network to connect to the internal network.
Internal network is used within the network.
Both do the same things the only exception is that they are used in different topology.
In every router topology, a firewall is needed to secure the network. The firewall allows the user to make certain rule to decide what type of traffic would be allowed in or out of the user's private network. Some firewall can restrict only certain IP address or domain names and some block certain type of traffic by the port they use.
Different organizations use different types of topology in their network.
 For instance, a small business uses a standalone perimeter router topology as it doesn't require a lot of security in it's network. This topology, connects the untrusted network to the company's private network through  perimeter router. 
Another example is the perimeter router with integrated firewall. This is more commonly use in small/medium organizations for a better protection of the network than just only the router protecting the network, and have a better rate of data transfer.
One more example is the perimeter router, internal router and firewall topology. This topology used both the perimeter and internal router. The perimeter router would connect to the firewall which would connect to the internal router which connects the public network to the private network of the organization. Usually large/medium organizations use this topology for greater routing options, better performance and protection.