Did I miss where a Rat and a Distortion+ are the same pedal?
It was never discussed but I can bring you up to speed.
As we can see here they are pretty similar circuits in that they use an op amp to provide gain for a hard clipped circuit. The big difference in sound we get is that the Rat has a pre-clipping high pass filter at 1.5kHz while the Dist+ is at 700Hz. The opamps produce similar amounts of gain, but I don't recall the values. The Rat has a mid hump at 1kHz the Dist+ is at 1.5kHz. For reference a Tubescreamer mid hump is 720 Hz. The Rat clips to 1.4Vpp and the Dist+ clips to 0.7Vpp, meaning that the Dist+ can clip at a lower input setting, but both pedals are capable of saturating the diodes with even the quietest of single coils. The Rat has a passive tone control that can go from a corner frequency of 32kHz down to 475Hz, while the Dist+ has a hardwired 16kHz treble cut. While 16kHz beyond the range of what most people can hear, it can effect the way the signal hits a later circuit further down the line. These are the major things but there are quite a few differences in pre-gain filtering values as well. The way the gain control works is slightly different between the two pedals, but not in a way that will affect the sound. The C4 capacitor in the feedback loop of the Rat helps to smoothen out higher frequency distortion. I don't know of the Dist+, but a Rat is capable of being used as a switchable buffer because with gain and filter at minimum and volume at max it doesn't color the sound at all.
All-in-all, if you look at the schematics, both of these op-amp based distortions operate in a similar manner, the key difference being in component tunings. You can very easily modify one to sound near indistinguishable from the other, minus the tone control. If you change C3 on a Dist+ to 22nF and replace the diodes with silicon, it should get you close. Remove C6 and R5 on a Rat, replace C5 with a 4.7uF, and change the clipping diodes for germanium and you should also be in the ball park. Might be worth experimenting with raising, lowering, or removing C4 while you're in there. As for the polka dots, paint will likely adhere to the mylar surface of the Rat better than the glossy surface of a Dist+ unless you are willing to get into sanding.
As a free bonus piece of info, if you know your opinion on the Rat2, Turbo Rat, and YouDirtyRat; that opinion will also apply to a Dist+; stock is closest to a YouDirtyRat, silicon will give you a Rat2, and red LEDs will give you a Dist+ that behaves close to a Turbo Rat.