One of the best-known differences involves the presidency. A term limit is built in for the Confederate president: one term of six years. It's a suggestion that's been kicking around as the basis for an amendment to the US Constitution for years, but nothing has every come of it. The vice presidency under the CSA Constitution is also a six-year term but, curiously, there's no limit to the number of times an individual could hold that office. Perhaps the drafters thought that no one would ever want to be vice president more than once, and if they did, so what?
Had the CSA survived longer than it did, Jefferson Davis' term would presumably have lasted until March 4, 1868. March 4 is mentioned in the CSA Constitution as the day the vice president-elect will start acting as president if no president has qualified by then, just as it is in the 12th amendment to the US Constitution. Davis had been elected in November 1861 with his inauguration on February 22, 1862.
Otherwise the presidency of the CSA, in constitutional terms anyway, is virtually the same as in the USA, down to the specification that a president must be at least 35. Even the Electoral College is replicated as method of selecting a president, with the House of Representatives as the plan B in case no one gets a majority.