I did some experimentation today with a pair of identical Rats (both are Biyang "Mice") with LM308N chips.
First, I confirmed that I had them set to get an identical sound.
After that, I subbed a OP07 into one. Despite my observations from the past, I was hoping not to observe a difference in characteristics. But lo and behold, the OP07 is a little brighter and more brash, and I had to back off the gain just a touch.
...After looking at the data sheets for both chips, all "standard measure" (i.e things that don't really matter much to pedals) specs were similar.
The one glaring difference is the LM308N has a large signal voltage gain of 300, the OP07 has a large signal voltage gain of 400.
...So there is about as close to the "smoking gun" as you are going to get, as to why the chips definitely behave/sound differently.
But I couldn't stop there...
I subbed in a LM301N, and tested again. Lo and behold, I could imagine many applications where someone may prefer this chip! It has MUCH better note detail, but the blur/growl of the LM308N is GREATLY diminished.
So looking at the data sheets again, I discovered the following differences:
input resistance - 500K to 2Meg
large signal voltage gain - 160
slew rate - 10V/uSec
input resistance - 10Meg to 40Meg
large signal voltage gain - 300
slew rate - .15V/uSec
The much higher slew rate of the LM301N gives it much better performance for fast picking and dynamic chord playing - lots more note detail. The 308 will be "blurry and tube-y" in comparison.
At half the large signal voltage gain, the 301 is going to be more like a 12AY7 to the 308 being a 12AX7. So it will never be as aggressive.
The much lower input resistance of the 301 (by many factors) will give it a softer response. But since the slew rate is about 100X improved vs. the 308, it will give a nice balance of "definition versus softness."
The other good news? The LM301 is about 70 cents versus the OP07's starting price of around 80 cents (both chips are expensive compared to something like a 4558).
In a lot of ways, the LM301 is a GREAT choice for folks who will use a Rat for lower gain chores - I could see it being very popular for playing blues and such where the 308 (or OP07) are just too growly and gain-y in comparison.
...So there are the results of my semi- to non-scientific experiments. All 3 of these chips DO sound different, and there are factual reasons as to why that is the case.