If they're kissing cousins then all of eschatology is part of a family tree that doesn't fork. The differences between the two views is quite marked.
Post-millennialism believes in the gradual bettering of society/world until it comes to a place where it is prepared to receive the return of Christ. That while the world exists in the state of growing grace it is the picture of the millennial reign of Christ. Christ returns after this for the consummation of the age. Post-millennialism was the predominate view of western Christians from about 1850 until about 1915.
Amillennialism is the believe that the age of mankind will come to an end at the return of Christ, there will be no millennial reign, and the final judgment will take place immediately upon His return. The New Jerusalem will be established and the wicked shall be cast away. Amillennialism is the predominate view of most academicians in theology but is not shared widely in the laity or pastorate.
When asked about these things I often reply: I am hopefully post-millennial (because wouldn't that be such a demonstration of grace), reservedly amillennial, but biblically premillennial.