I am a booked customer on a mid-May FI trip from CLE to AMS (economy class). Here's what some other airlines want versus what I paid.
CLE-XXX-AMS (RT)
AA $1306.36
DL $1307.36
UA $1307.36
FI $814.32
I'd say that's substantial savings.
FI's price includes the baggage fee and free selection of seats at booking, but food is extra. Baggage is extra on the other airlines (although it's waived for elites and certain CC holders) but food is free. So, there are some fee differences, but not too consequential.
Yes, definitely a difference. Status may be a factor, too, as some UA flyers in Cleveland can automatically get E+ which really has no comparison on FI (unless you consider a 33" seat pitch "comfort"). I don't think the baggage comments are correct, though. I know UA allows one free checked bag on transatlantic flights and probably the others do, too.
BTW, for comparison I just priced out a hypothetical CLE-AMS for next May in Saga Class and it came out a little over $2300.