Why do scientist now rely on the moment magnitude scale to measure earthquake magnitude?

The Moment Magnitude Scale (I'll call it MMS) is used to compare the amount of energy released by earthquakes. It's based around the seismic moment and the moment magnitude using a pretty complicated formula although it is designed to be comparable to the Richter Scale. Each step corresponds to 10 to the power of 1.5, so 2 steps is 10 to the power of 3 or 1000 times the increase in energy.

The main advantage between the MMS is that the upper end does not lose it's effect, i.e. there is no upper value that beyond which all earthquakes have the same magnitude. That is the reason for using it to measure larger quakes. It's only used for earthquakes with a magnitude of 3.5 and above.

With the Richter scale, the original Local Magnitude cannot be calculated for events great than 6.8 which led to possible extensions and finally the new Moment Magnitude Scale. The Richter Scale is based on old technology around in 1935 and it is hardly surprising that the scale is now largely outdated.

The Richter Scale was designed for earthquakes along the San Andreas fault only.

The richter scale is a measure of the amount of shaking experienced during an earthquake - not a measure of the actual amount of energy released during the quake - so it really doesn't tell you anything about the earthquake from a scientific perspective - shaking may vary depending on where the seismometer is located (more shaking would be experienced on sediments than on bedrock, for example - like if you compare a house built on Jelly to one on the table!) and also on the building within which the seismometer is held - the design, and how earthquake-proof it is, will affect the shaking experienced. Moment magnitude, on the other hand, is calculated from the amount of slip which has occured on the fault, and the area over which it has taken place (eg if 200 square metres of fault moved 10 metres or something) - and is therefore a much better measure of the actual size of the quake.  The richter scale is still quite useful when dealing with the general public, however, as the numbers produced from the moment magnitude measurement are rather large and scary looking, where as richter scale is just like "4.5" etc...
Hope that helps