What's the difference between EM HEAT and AUX HEAT?

the EM heat setting is the efficent setting uses less energy the aux uses more energy to heat quicker

Sorry, neither of these answers are completely correct.
You have a heat pump (or the wrong thermostat). Let's assume you have a heat pump.

In air conditioning mode, it works like every air conditioner you have ever had, but...

In heat mode, it reverses its operation. Have you ever felt the air coming out of the outdoor unit of your A/C unit? It's hot, isn't it. And the air coming out of the indoor unit (out of the registers) is cold. Now for a heat pump to produce heat it simply runs the air conditioner in reverse and the heat comes out in the house and the cold is released outside. Neat, huh!

Here's the problem with heat pumps...when it is really cold outside the heat pump can't produce enough heat to heat your home. So it has an additional heat source called "Auxiliary Heat". This heat comes on automatically when the house doesn't get warm enough. The source of this heat is based on the region of the country you are in. North/Northeast generally have oil heat, other regions have gas, and still others have to use electricity to heat. In Texas, we usually use electricity as the supplementary heat on heat pumps. VERY EXPENSIVE!

Now the "Emergency Heat"...this is exactly as stated in Solution #2. This is manually turned on by YOU at the thermostat when your heat pump fails. This turns on the auxilliary heaters and turns off the heat pump (remember, the reverse air conditioner). Again, this can be quite expensive to run if your heat source uses electricity! Gas and oil may be cheaper. The emergency heat is only designed (normally) to keep the house livable (not comfortable) until the Heating Tech can get out to you and fix your heat pump.

Something else you should know. It is normal for a heat pump's outdoor coil to frost up during heating mode. It will detect this and go into DEFROST mode and melt the frost off the coil. While it is doing this, it will turn on the auxilliary heater to keep the air blowing in the house at a reasonable "warm" temperature, but it will not be as hot as normal. In fact, heat pumps produce a lower temperature heat than traditional heaters. So the air may feel cooler during heating than you are use to anyway. This is normal and is not a sign of a problem.

So what do you do: Set your thermostat to the temperature you want and set the controls for HEAT/COOL and FAN-AUTO/ON and leave the EMERG HEAT off unless your heat pump breaks.

As always, keep your filters clean and your outdoor unit's coils clean and free of debris.

Hope this explains your question for you!

sounds like you have a heat pump. Emergency heat is usually used if
mechanical failure of heat pump occurs. This is usually 2nd stage heat
which can be oil fired, gas fired, or electric. This switch bypasses 1st
stage and uses strictly 2nd stage heat. Auxillary heat is generally
related to supplemental heat which does not always have to be part of
the heat pump. It can be perimeter baseboard or even radiant heat
supplied by a boiler. These are just examples. Please review all
equipment within your home. Hope this answers your question.