What's the significance and difference of a farewell gateway?

The difference of farewell gateway is - GATEWAY A very simple answer would be, a gateway is a node on your network that acts like an interface to other networks. Typically this is the machine that the desktops and workstations connect through to access the internet. A better answer would be that it allows disparate networks to communicate with each other. At its heart a gateway is a machine that can translate the protocols used on one side of the network with those used on the other side. Before the widespread adoption of TCP/IP in the early days, gateways were the network "masters" that talked to each other, they were known as IMP's (Interface Message Processor) and were designed so that instead of every computer having to know all the protocols in the entire ARPANET the IMP would act as a gateway and know those used at a particular site and it's native protocol used to communicate with other IMP's.

Today gateway computers serve much the same purpose but instead of translating communication protocols they often function as DNS/DHCP servers and translate addresses taking the single IP that a site may own and then figuring out which computer to route that packet to. This is done by routers as well, many home routers have built in DHCP servers to allow you to connect multiple computers. Consumer routers may also perform common firewall tasks such as port forwarding, port blocking, and packet filtering.

FIREWALL A firewall is a software or hardware appliance that blocks, filters, or in some way restricts and to a degree directs network traffic. Typically the firewall would be used to selectively block connections to various ports on local machines that the admin didn't need. It may allow ssh connections to 22 but only from a certain IP block or it may not allow any traffic on ports commonly associated with P2P networks. They are much more advanced now, "next gen" firewalls can do application filtering, NAT, and proxy services. These though are mainly software firewall programs such as those created by Microsoft, Barracuda, and others. The hardware firewalls, specifically the Cisco ASA series, are very advanced intelligent appliances that can defend against and mitigate multiple threats. These are generally enterprise grade, many small and even some medium sized businesses won't find the expense necessary and will stick with older appliances or software based firewall applications.