The difference is in how the force/weight is measured. An RCI or rated capacity indicator uses load sensors that measure the force directly on the hoist line. For example, a dynamometer, also known as a "line rider" uses a load pin in a sheave that the hoist line exerts a force on. The pin then outputs an electronic signal, which can be translated into the weight reading.
The other common type of load sensor used with an RCI is a load link that measures the force being pulled on the hoist line. These can be installed on the "dead-end" where the hoist line is secured to the boom or when using a wireless load link installed above the hook block. Both methods involve measuring the force directly off the hoist line that is lifting the weight.
The LMI or load moment indicator uses sensors that measure the force on the boom, not directly on the hoist line to calculate the weight being lifted. An LMI can use load pins that measure the force on the pendant line on a lattice boom crane or pressure transducers that measure the rod and bore side pressure of a boom lift cylinder.
LMI systems use physics to arrive at the weight based on the force on the boom along with the boom angle and length.