The 31-year-old has been handed a four-game suspension and a £220,000 fine after being found guilty of racially abusing Queens Park Rangers defender Anton Ferdinand.
The English governing body claimed that the former England international had been given a reduced ban than that handed to Liverpool striker Luis Suarez - who was banned for eight matches for the same offence - because Terry had not repeatedly used the word "black".
"It may be when you look at all the detail they thought there were reasons for [the shorter suspension]. I can't see it," said Triesman.
"He's within his rights to appeal. My own view is it would be more sensible to apologise. I don't believe in this day and age that anybody can think that it's okay, and that you don't owe an apology, not least to the other player."
Reports have claimed that Terry will not appeal the sanction.