What is the difference between a multimeter with RMS and one with True RMS?

The answer to this is a solid "it depends". More specifically, it depends what kind of signals you are going to apply the meter to.

If every AC signal that you wish to measure the RMS value of is a pure sine wave, then you don't need a true RMS meter. If, however, you want to measure the RMS value of a square wave, the output of a half-wave rectifier or something else more complex, then a true RMS meter will be advantageous. An example of where this might be relevant is if you are trying to calculate the power dissipation of a resistive load in an AC power system where the mains has been through some sort of processing, or perhaps if it is being driven with a PWM signal.

Although it is a load of marketing, Fluke's website has a good article on the matter here. One nice figure it gives is that for a square wave, a non-true RMS meter will read 10% high when measuring the RMS value of a square wave (and this will vary by pulse width for a PWM signal).

By the way, Dave Jones over at EEVBlog did a $50 multimeter shootout a few years back. A bit outdated, but still useful for reasons to choose a particular meter without tRMS and then the $100 multimeter shootout covering tRMS meters.

There are a number of different types of AC-voltage measurement (peak-to-peak, RMS, etc.), and they'll generally yield different values for any given signal. In many cases, if one has a measurement of a known type and one also knows the shape of the waveform and DC offset (if any), it will be possible to compute what the other measurements would have been (e.g. for a sinusoidal signal with zero offset, the peak voltage will be about 1.414 times the RMS voltage), but a number by itself, without information about what kind of measurement it represents, is apt to be meaningless.

For many purposes, sinusoidal waveforms are reported as RMS voltage (a 120V or 240V power main, for example, will nominally have 120V RMS or 240V RMS), but cheap meters will often measure AC voltage via some other means and then scale the result in whatever fashion would be appropriate for a sinusoidal signal with zero offset.

If one is measuring a sinusoidal signal with zero offset, such a meter will work just fine. In other cases such a meter may still be usable (and in fact may sometimes be better than a true-RMS meter) if one knows how its measurements are computed and can figure out from that what one wants to know about the signal (e.g. if one has a meter that is known to measure the peak voltage and scales it by 70.7%, and one wants to know the peak voltage of an irregular signal, one could use such a meter by multiplying its displayed result by 1.414, while an RMS meter may be nearly useless).

The primary advantage of a true RMS meter is that it will measure irregular waveforms in a known fashion, subject to documented frequency restrictions. Other kinds of meters may perform measurements in ways that would be sometimes more useful and sometimes less useful, but unless the meter documents the actual measurement techniques used, they're apt not to be useful at all.

An inexpensive multimeter measures the average of the full-wave rectified AC voltage and fudges the reading upward by a factor of:

\$\frac{\pi}{\sqrt{8}} \approx 1.111\$

to match the RMS value of a pure sine wave.

This means the reading will be considerably in error if you want the RMS (heating value) of something like a low duty cycle pulse (a large crest factor). The average will also not be (directly) displayed, but you can divide by 1.111 to get it.

Circuits that perform 'true RMS' calculation have maximum bandwidths and dynamic ranges, but within that range they can do okay, say for things like measuring the RMS voltage from a phase controlled dimmer. They do tend to cost more, and have more error than the simple averaging circuit.

If you are doing mains voltage work you should consider a true RMS meter that is appropriately safety rated as well. For most electronics, work it's really not required, and if you need to look deeper, you should save the money and get a good oscilloscope, which will tell you a lot more.