The answer to this question is fairly simple. The Oade Concert Mod 661 replaces the stock bipolar transistors used for gain with better performing chips, but the same type. Bipolar transistors are considered "transparent".
The Oade Warm Mod 661 replaces the stock bipolar transistors with FET chips, which are more valve like in behavior and sound, producing the warm tone that is familiar in older recording equipment that used FETs as a general rule in gain staging.
So, what does this mean? Doug gave me a reference point when I was trying to decide, and I have relayed this information in the 661 threads that you may want to search for and read, so you can get answers to questions already asked on this forum, without asking them again.
He said that the Concert Mod 661 was similar in tone to his M248 preamp, and the Warm Mod 661 was similar in tone to his M148 preamp.
That being said, people that run mics that are typically dark sounding (Schoeps) or are flat or bass heavy seem to find that the M248, with it's shimmery and transparent sound, counters these qualities and rounds out the sound of these types of mics.
People that run mics that are typically leaning towards being bright (AKGs) seem to like the enhanced warmth in the lows and mids, along with the smoothing out of the high end energy that the M148 delivers. This was why I picked the Warm Mod for my 460s.
So, if you are intending to run Schoeps, my advice is to get the Concert Mod, and I expect if you posed that question to Doug directly, he would agree. The warm mod will probably be way too bass heavy running Schoeps.
Good luck !!!