If I had neither, I'd go for the Passport instead of the Classic in a second. The Classic is large but fragile. The Passport has those great off-white patches for setting WB. You get a larger white and gray card too. All protected in a case.
The problem with the regular ColorChecker (even the small one that is discontinued I think) is that it's very delicate and has a short shelf life (they will only last several years at best). The Passport has a hard outer case that protects the CC card as well as offered additional targets such as warming/cooling WB as well as color samples that are custom tuned to match up with the 8 hue colors in Camera Raw/Lightroom's HSL sliders.
Seriously, if you want to have a handy target for making profiles, you would be an idiot to buy the full large CC card and ignore the other bennies of the Passport.
The key thing you need to understand regarding the differences between using the Adobe Standard DNG profile and making your own custom DNG profile is that it's quite possible that YOUR sensor (the one in YOUR camera) is slightly different in it's spectral response from the camera/sensor Adobe used to create Adobe Standard. Therefore the custom DNG profile may be more accurate for YOUR camera...
Bruce Fraser and I tested two cameras (the original Digital Rebel) and found substantial differences between the two lots of cameras...we had different spectral responses and different native ISOs. His camera was about 1/3 stop hotter than mine and our cameras needed slightly different camera calibration settings.
Each sensor lot is bound to have individual and slightly different responses...if you value the best potential capture, you would do well to do a custom profile of your camera's sensor. The same capture can be used in the Passport software and the free DNG Profile Editor...if you are anal, you can try creating a profile in either–both will use the same targets (full size CC or Passport). Note that the DNG Profile Editor allows further editing of the resulting profile.
Also, be sure you understand the purpose of a DNG Profile...it's not a color editing tool to take the place of properly adjusting the image colors, it's a color rendering tool to allow you to start off with an optimal color rendering that you can then adjust further.
One is a technical tool and the other is the color correction tool...DNG Profile for technical, the ACR/LR sliders for color correction.