If your server is acting as the firewall (NAT, Squid Proxy, etc.), then having an internal subnet is important too. When doing this, however, it's also best to place the public (external) and private (internal) networks on separate physical NICs.
Setting up different IP address netblocks should not make anything noticeably faster or slower, but it will require that some care go into planning how you're going to organize your network. For starters, on the server you'll only need to set the gateway to match the one that your ISP provided for your public Static IP address (NAT, proxy, etc., will use this on behalf of requests from internal computers).
The beauty of using private IP addresses is that you can have many computers, devices, etc., connected to the internet with a single IP address. In fact, this is a very common setup with NAT, and one that occurs with broadband/wireless routers (e.g., what LinkSys sells en masse).
Some helpful links:
RFC1918 - Address Allocation for Private Internets (see section 3 on page 4)
Step-by-step configuration instructions for NAT with IPTables
A discussion in a forum about setting up what you want to do
How to Install and Configure DHCP Server in Ubuntu Server