Characteristics of DMPL and Proof MorgansDMPL Morgans were struck for circulation, so they will have the characteristics of a business strike coin. For DMPL Morgan Dollars, this means that there should be some bag marks and dings and scuffs, despite the beautiful mirrored surfaces. Proof coins, on the other hand, were made from specially prepared planchets, and struck with dies that were carefully finished and polished. Proof dies were not used very long, so the sharpness of the strike should be superior, plus Proofs are struck with greater striking force with multiple strikes of the die faces. In addition, Proof Morgan Dollars shouldn't have seen any circulation, and were certainly never transported in canvas bags of a thousand coins at a time, all rubbing and jangling together like the business strikes were.
Telling the DMPL Apart From the ProofProbably the easiest and most obvious way to tell the DMPL Morgan Dollar apart from the Proof is to look for signs of circulation. Since Proofs didn't circulate, and shouldn't have come into heavy contact with other coins, Proof coins will lack the little dings, scuffs, and marks that circulation (DMPL) coins have. If the DMPL is especially clean, an unscrupulous person, or even just a novice, might try to sell it as a Proof, which is why my first answer to this question was to buy a slabbed coin.
A Comparison View of DMPL Versus ProofFor someone who has a practiced eye, the Proof surfaces are clearly different; they almost jump right out at you. For someone less experienced, perhaps the most surefire way to tell the very clean DMPL apart from the Proof is by using a comparison coin. Compare the subject specimen to a known Proof Morgan, preferably from the same mint and year, (although if you can only have one or the other, take a Proof from the same Mint.) Examine the surface quality under a 10x loupe or microscope. Does the subject specimen match the Proof in texture and depth of sheen? Look for evidence of metal flow lines on the subject; Proof coins rarely have these, since flow lines stem from die wear and Proof dies aren't used for very long.
If you don't have a Proof Morgan Dollar to compare to, keep in mind that another key point of differentiation between Proof Morgan Dollars and DMPL Morgans is the sharpness of the devices. Because the Proof dies weren't used very long, the devices, especially the letters, numbers, and stars should be very sharp at the edges, and rise straight up from the field when viewed under good under magnification. Business strike Morgans will have a softer, more rounded, and less clearly defined "presence" from the field, even the coins which were struck from a fresh pair of dies. This is because the Proof gets at least two strikes onto specially prepared planchets, whereas the business strike coin got one strike on a random planchet.