Two things to start out with testing:
1) Is the problem computer farther from the router and/or does it have something in between it and the router like a central heating blower or microwave? The symptoms you're describing are pretty much the behavior if the signal is very weak. If you can, try moving it closer and see if it works properly; if so, there's your problem.
2) Try selecting "Turn Airport Off" from the drop-down Airport menu, then immediately selecting "Turn Airport On". If it then connects automatically and properly, you're seeing a the same problem I've had intermittendly for years--the Mac somehow fails to automatically connect (usually after wake from sleep) until the hardware is toggled off and back on. If this is your issue, I've never found a workaround and have been goofing with it for years. I assume it's a hardware problem.