Exported AIFF are NOT lossless?

Question

When I perform a "Share...Songs to iTunes" in AIFF format, I can DEFINITELY hear a difference between the version played with Logic Pro X, and the saved, supposedly lossless, AIFF filed played with the iTunes player.
I notice the same audio difference when this song is saved out of GarageBand as an AIFF file.
Is this a known issue? This would put into question the integrity of the Apple Platform to reproduce pure, unaltered and error-free recordings.
Does the problem reside instead just with the iTunes player (not likely) ?

Answer 1

"Offline uses available CPU and other resources  to the max..."
Why?  It doesn't have to do anything different than what it can do realtime.  I think most (99%) people would agree that they just want the bounced or saved mix to sound exactly the same as the way they hear it through the headphones.  They would rather have it bounced (and/or saved to file), without ANY alteration, rather than hope a 3rd party plug-in somehow get its 'optimum quality settings engaged' so that the mix turns out even better than what they listened to prior to bouncing (or saving).  So if a workstation has enough processing power to accurately preserve a mix in realtime, then it certainly should be able to do the same thing offline (only faster).
I'm thinking this is a deficiency (bug) with the way Apple is bouncing/saving their mixes out of GarageBand and Logic.  It is amazing no one has brought this serious issue up before. 
Possibly in the past, this issue has provided a convenient means to scapegoat third-party plug-ins for altering (screwing up) the sound of a mix?  But I wasn't using third-party plugins in my mix, so the problem obviously resides with Apple software.
As a minimum, since offline bouncing, and sharing files (to iTunes) can indeed alter the sound of a mix, Apple should at least provide a setting to bounce all mixes down with the same sound preservation that is being achieved currently with only their real-time bouncing.  (AND IT SHOULD BE THE DEFAULT SETTING, PARTICULARLY FOR GARAGEBAND USERS THAT DON'T HAVE THE OPTION TO PERFORM REALTIME BOUNCING!  DUH!!!)
[Note: I'm not SHOUTING at you, 'The Art of Sound'; you have been very helpful.  I'm hoping somehow, people reading this message, including APPLE, will all get on the same page and agree that the way to do things, when bouncing or saving files, is to PRESERVE THE MIX (short of compression loss) as best as possible for the default setting.