Both are measures of instability and can be useful in determining thunderstorm potential, but there are some differences.
LI is the lifted index which measures the difference in temperature between a parcel of air at a certain height (500 mb, for example) and the actual temperature measured at that height. If that parcel of air is significantly warmer than the actual temperature and surrounding environment, the lifted index is negative, and the air parcel will continue to rise until it reaches a level of equilibrium with the environment. That represents unstable conditions, and the lower the number, the more unstable, meaning a greater temperature difference and more instability at that level of the atmosphere. As numbers move more positive, the atmosphere is more stable, and air parcels will not lift significantly from their currently level. If air parcels can't be lifted to greater and greater heights, thunderstorms will have difficulty in developing and becoming strong/severe.
CAPE is the Convective Available Potential Energy. Its generally a better way to determine instability, but the key word is "Potential", so the measurement is just possible total energy that can be used by a parcel of air, not necessarily how much will be or is in use at that moment, as other factors (such as a Cap or CIN - Convective Inhibition) will have a negative feedback on energy. Like the lifted index, you can look at CAPE at multiple layers so the CAPE may be high near the surface but much lower higher in the atmosphere, which will impact storm and updraft strength. Large, tall areas of High CAPE will mean greater updraft strength and potential severity in a storm.
