History
The dial-up modem is what made the Internet into a booming success in the '90s. From a technological perspective, the dial-up modem is however severely outdated. It was designed for a young and simple Internet.
Benefits
The dial-up modem lives on due to its still much wider accessibility than its modern counterparts since it doesn't need a subscription with an Internet service provider to function.
Comparison
The PS3 can play films and games in full HD resolution. A typical dial-up modem can't stream neither low-quality audio nor video.
Speculation
With a data-rate of 56 Kbit/s and a latency (the time it takes to send the data back and forth) of 400 ms, it would be plain impossible to play most PS3 games online and would take 54 hours to download a typical-sized 1200 MB demo of a game. That's 3,240 minutes of connection that costs an average of two cents per minute, which equals to $64.80--twice the cost of a month with a typical 8 Mbit/s DSL/Cable connection.
Considerations
To use the popular online features of a PlayStation 3 it would be wise to at least use an Internet connection with a set, monthly fee. 768 Kbit/s is a recommended minimum.