I like both "Maine" and "Louisiana". "Maine" has John's classic unreliable, evil narrator, while "Louisiana" just feels like a fun take on a hostage situation.
"The Songs of the 50 States" is also dear in my heart as it gets gradually crazier.
I'd say I'm generally neutral between Flans and Linnell. I might say that Linnell's songs tend to be of consistent quality, but when Flans puts out a good song, it's a great song. Some songs are just better with Flans's helium voice.
Edit: To clarify, I like the experimentalism of Flans's stuff. I find the band needs a balance of both.
I do enjoy both of Linnell's albums/EPs a lot, though. I haven't listened to much of Flansburgh's non-TMBG stuff like Monopuff.