Hello Mahmoud.
These two commands do two very different things. First of all, the logging monitor errors command, does enable error level logging as you mentioned. That is, it logs all level 3 messages and below (Errors, Critical, Alerts and Emergencies). This command places all syslog messages into the local logging buffer (or sends them to the syslog server, depending on the configuration).
Conversely, the terminal monitor and the terminal no monitor commands don’t turn logging on and off. These commands indicate wheather or not the logging messages will be displayed on the command line interface.
By default, when connecting via the console, logging messages are displayed on the command line. That is, terminal monitor is the default setting. When connecting via Telnet, logging messages are NOT displayed by default, that is, terminal no monitor is the default setting.
So when connecting via Telnet, if you want to view the syslog messages on the command line as they occur, you must type terminal monitor. Note that even if the terminal no monitor command is used, syslog messages are continually being recorded, and you can see these using the show logging command.
I hope this has been helpful!
Laz