They accomplish many of the same things with both allowing you to state who gets your assets when you pass away and naming guardians for your minor children.
A last will may be subject to the probate process, in which a court reviews the will and orders the distribution of your estate according to the will. Depending on where you live and the value of your estate, probate can be an expensive and time consuming process, or relatively quick and inexpensive.
A living trust avoids the probate process if you transfer your assets, such as homes and financial accounts, into it before passing. This process takes significant time and effort, but can make the process of administering the estate much easier when the time comes.