Does anyone know the difference between parts in the title?

I think both are John Cooper Works Lower Grille Trim Black Chrome.

I have 2012 MC base, production 12/2011. Purchased and installed 51130417137, however, it does not line up with the upper trim. The lower trim stick out about an inch. I compared it to the original chrome part and it is little wider.

 

7137 is listed for Cooper and One models, 2011 is listed for Cooper S. Both are listed for cars made up to August of 2010. I wouldn't expect either part to fit your 2012 Cooper. Realoem.com lists different part numbers for pre-LCI ('07-10) and LCI ('11+) models for the chrome lower trim, so clearly it changed. OutMotoring lists the 7137 part as fitting all Cooper models regardless of year, but I suspect this is incorrect based on the Realoem.com information and the fact that the grilles themselves are different on pre-LCI and LCI Coopers.

My guess would be that, like the JCW grille itself, they don't make JCW grille trim to fit a 2011+ Cooper (non-S), but I could be wrong.