Cat5e is good for 1Gbps max. Cat6 will do 10Gbps. If you're pulling cable through walls, Cat6 will be more "future proof," however Cat5e has plenty of life left. Google "cat5 vs cat6" for tons of info.
The "advanced router" has 1Gbps wired Ethernet ports while the older router is 100Mbps only. The new router has 802.11n vs. 802.11g on the old router. There are some other improvements, but these two are the big differences for most users.
Have Fun.