I've found the problem. Lately I've changed my WiFi network's security from WEP to WPA2. In
Network Preferences | Advanced | Preferred Networks, my network's name was still figuring with "WEP" under Security. That's why it wasn't connecting automatically.
Removing the network from the
Preferred Networks list, disconnecting and reconnecting (so that my network would be re-added to Preferred Networks, this time with "WPA2" security) solved the problem.
BTW, I kept using the same passphrase for WPA2 as for WEP, so the information in the Keychain remained valid, and that's why manually connecting worked without prompts. Apparently in the keychain, the passphrase is only tied to the network name (SSID), while preferred networks are tied to both SSID and security type.