Which is more beneficial, a router and a firewall or a router with a built in firewall?

You can go for any of the options depending upon your requirement. Most of the Cisco routers are available with the software firewall and IOS options. You can get any router with firewall and/or IDS IOS feature set.

A router with in-built firewall will be economical option but then you will loose on processing power of the router, you will not get a statefull transition in case of failure and you might have to consider interoperability of different features you configure on the router.

A router with a separate firewall will do all these - but will be costlier. Just remember, routers are meant for routing not filtering and firewalls are meant for filtering.

IDS is a host which just collects all sessions and analyzes them for illegal or unauthorized connections and will then report about them.