Why don't typical digital multimeters measure inductance?
The only reason DMMs can't measure inductances is that it is more difficult to measure inductance than resistance or capacitance: this task requires special circuitry, which is not cheap. Since there are relatively few occasions when inductance measurements are required, standard DMMs do not have this functionality, which allows for lower cost.

Simple DMMs can measure capacitance by just charging the capacitor with a constant current and measuring the rate of voltage build-up. This simple technique provides surprisingly good accuracy and wide dynamic range, therefore it can be implemented in almost any DMM, without significant cost penalties. There are other techniques as well.

Theoretically, one could measure inductance by applying a constant voltage across an inductor and measuring the current build-up; however, in practice this technique is much more complicated to implement, and the accuracy is not that good as for capacitors due to the following reasons:


Inductors may have relatively high parasitic resistance and capacitance
Core losses (in cored inductors)
EMI (incl. stray inductance and capacitance)
Frequency dependent effects in inductors
More


There are few techniques for measuring inductances (some of them are described here). 

LCRs are special meters designed for inductance measurements and containing the required circuitry. These are costly tools.

Since the hardware for measuring the inductance may also be used for accurate measurement of R and C, LCRs also employ this circuitry in order to improve the accuracy of capacitance and resistance measurements (for example: AC resistance, AC capacitance, ESR etc.). I believe that the difference between measuring inductance and capacitance with LCR is just a matter of different firmware algorithms, though it is just a guess.

Therefore, the general answer to your question is "yes, LCRs are usually more accurate in RC measurements than DMMs, and they can measure a wider range of measurable quantities". However, this is just a rule of thumb - there are many superb DMMs and lousy LCRs out there... Read specs.