The Local Government Center (LGC) is a program of the University of Wisconsin Extension created in 1992. It's mission is to provide focus, coordination, and leadership to UW System educational programs supporting local government, and to expand the research and knowledge base for local government education. In short, they take the knowledge of the University of Wisconsin System, format it in various ways to meet the needs of local government and deliver it as training and education programs to local government officials and citizens interested in government.
The Local Government Institute (LGI), despite the similar sounding name, is quite different. It is not affiliated with an educational institution. It is affiliated with the associations representing local governments in Wisconsin - the same associations that promote and support LGC training opportunities. The main focus of the LGI is working with others to find solutions for the efficient delivery and funding of local government services.
LGC is one of several groups the LGI partners with to fulfill its mission. Occasionally LGI will put on a workshop, host a conference or a webinar or otherwise conduct a training event. Sometimes these are done in conjunction with the LGC and sometimes not. Any training that LGI conducts is specifically focused on local government collaboration for the efficient delivery of services.