Usually, but not always, proof dies are identifiable by the way the dies themselves were finished. The fields on proof coins are normally polished and will lack any fine lines from grinding when viewed under magnification. Proof dies are POLISHED to remove fine grinding lines and any other stray contact marks. It is also customary in some countries (like the US) to put a slightly different finish on the recessed portions of a proof die. The raised details on the coin in that case come out frosty. The first strikes from a new die pair will look the best. The frosty surface wears off quickly because of metal flow on the surface. The preparation of the proof die actually involves a slight mechanical or chemical erosion of the surface to produce the "matte" effect.
Some proof coins will also have a slightly different appearance at the edge of the coin - the striking pressure is higher for proofs and the impression is deeper resulting in a higher often wire like rim. Some proofs feature a wider collar to create a nicer edge detail.
All of the features may or may not be present on any proof and at times the clues are very indistinct. You really need to know the characteristics of the mint involved. If the coin has "finger" lines caused by the mechanical fingers that place and remove the coin - it is a business strike.
As I noted above proof dies are also used at a higher operating pressure on presses that run at much lower speeds and often strike a planchet multiple times. Two strikes are standard for US proofs. This means the dies wear quicker (producing fewer coins) before retirement. Until recently many mints used "worn out" proof dies to make business strike coins. If the dies started out identical (no changes in the design for proofs) - then the products can not be distinguished once in circulation.
But at times, proof dies DO have a slightly different design from the business strike dies. This has been done at various times to sharpen certain features of a coin for collector sets. US quarter dollars struck in the 1960s are a classic case. When this happens you can often ID coins made from old proof dies that were used to make business strikes before their final retirement.