Some will, some won’t as much between them. The opamp does clip in these circuits as well, but never before the diodes do which gives full on clipping even with very low gain setting. There are a couple of things that will make a bigger difference between them though.
The first is the input coupling cap combined with the loading the guitar sees. The MXR will let more low frequency info through, and will load the guitar down less. It also is fed at a much higher impedance from its bias rail. The DOD cuts a bit more bass (even more in the yellow 80’s version and the yjm308) and also loads the guitar down a little more.
The second is the .001uf cap to ground right at the input of the MXR. That is going to directly affect the resonance frequency of the pickup and push it down. It is akin to having a really long or cheap cable.
The third is that both circuits have a similar shelving filter within the -fb loop surrounding the opamp (lowpass shelving filter comprised of the feedback resistor 1M, the .047uf capacitor, and the combination of the 4.7k resistor with the gain pot. Because we are talking about -fb and in a perfect world 180deg inversion - it’s not though, any capacitance with in this loop shifts that - it manifests as a high pass shelving filter as far as the output is concerned.) this makes the amplifier gain frequency dependent. At the max setting of the gain, both are exactly the same giving a highpass shelving boost with a -3db point at 720hz near the filters peak frequency, starting its 47db rise at 3hz. The difference between the MXR and DOD is when you start rolling the gain down.
The fourth is at the output. The MXR and the reissue Yellow 250 from the late 90’s early 2000’s will only give you a maximum 250mv - 350mv output signal from the device. This may have been done on the non germanium dist+ to more closely mimic the expected output signal from it. The older 250’s and the YJM308 will allow the full signal level set by the diode turn on voltage, through - a heavily clipped 500mv-700mv.