I can hardly understand your situation.
I’d like to clarify what you faced.
You can send data from serial part in serial command mode but can’t do it in data mode, right?
If so, there is something strange or mysterious.
In fact, most of our serial to ethernet module including WIZ105SR can’t communicate to a peer system in serial command mode but those can do it only in data mode.
You said you could do it inversely.
Could you write your system environment and test scenario in detail as a reply?
I may be able to analyse what happened.