There are two main differences between the LCMS and the WELS.
One is fellowship and what is necessary to have full fellowship. The two used to have full fellowship until 1961. At that point both sides seemed to have changed some of their fellowship criterium; the LCMS is less stringent in certain aspects regarding fellowship, and the WELS became a bit more stringent on theirs because they viewed that the LCMS was sliding in the other direction. These moves were enough to cause fellowship to be broken.
The other has to do with church ministry. The LCMS views the Office of the Pastoral Ministry as the one divinely instituted office and applies the Scriptural teachings regarding the ministry (e.g. who can and cannot be called) to that Office and not necessarily to man-made offices in the church. The WELS applies the Scriptural teachings of the Pastoral Office across the board to include man-made offices as well. They view all offices in the church as a part of the ministerial office. The LCMS, however, does apply those Scriptural teachings to those positions that are directly tied to the functions of the Pastoral Office (although, unfortunately, not all congregations follow this).