I think most people will say that the JHP has a slight accuracy edge in most guns over the CMJ. Whether that is because the mass is located more "outward" along the rotating axis on the JHPs, the base of the JHPs are more uniform or consistent, or if it is even true in most guns is up to debate. I have shot both and don't remember there being a major difference, if any, but I also don't remember doing any real thorough testing of them. Both were easily adequate enough for my limited division gun in competition. I have a case of MG, either CMJ or FMJ, I can't remember, that I will get to eventually. The only reason I don't shoot MG all the time is I can get other good bullets a little cheaper.
The advantages of the CMJ that I can think of would be slightly lower cost, the ability to tumble loaded ammo and not get media in the nose (not necessary and I don't tumble my ammo so this is a non-factor for me personally), and I guess potentially a small edge in feeding. I would be surprised if this was actually the case though, as the nose of the MG JHP is a very rounded profile and feeds very well. If the gun and mags are close to right and the ammo is loaded within a reasonable length range for the gun, these will both feed 100% in my experience.
If you were looking for a pure accuracy load, although you could try the MG's, I would be willing to bet a Sierra, Nosler, or Hornady bullet would beat both in most applications, albeit at a higher cost for most likely a small benefit. Zero also makes a competitively priced JHP that shoots very well, especially for the cost.
www.rozedist.com I shot these for a while several years ago and still have a couple thousand around. The only reason I don't shoot these is also because I can get other decent bullets a little cheaper.
I know this is a bit beyond what you asked, but I've been shooting Precision Delta FMJ's for quite a while and they are fine for me.
http://precisiondelta.com/ They are an exposed lead base and give up a little to the "best" bullets in pure accuracy, but they are an acceptable tradeoff of cost/performance to me, at least for USPSA/IPSC.
My $.02 anyway.