And, could you please give me examples or cases in which we can't use the other word?
The following can't use "transport":
We're transferring you to the South branch. (A job transfer)
I want to transfer $10,000 into a term deposit.
Surgeons can transfer a donor kidney from a cadaver into a sick patient.
You're transferring your feelings about your father onto me!
The following can't use "transfer":
Driver, please transport me to the station.
Strollers and prams are used to transport babies.