What are the differences between the HMQC and HSQC experiments?
You get a better resolution in the indirect dimension with an HSQC compared to an HMQC. The HSQC signal is a singlet while the HMQC is usually a multiplet as you see certain couplings on it. So generally you should prefer the HSQC over the HMQC.

One advantage of the HMQC is that it uses significantly fewer pulses than the HSQC. This means that errors in the pulses don't accumulate as much as in the HSQC, you'll lose less signal if your pulses are wrong than in the HSQC.

If you had trouble setting up an HSQC, that could mean that the probe wasn't tuned and matched properly for your sample, or that your pulse lengths were wrong. The proton pulse should be determined on each sample individually; the $\ce{^13C}$ and $\ce{^15N}$ pulses should be set up correctly already. If you're not using a probe with automatic tuning and matching you should also do that manually before the measurement.

There are a lot more HSQC and HMQC variants that are used, e.g. Sofast-HMQC or TROSY sequences that are used for larger molecules like proteins. Another interesting variant you should consider is the multiplicity-edited HSQC, where $\ce{CH2}$ signals will have the opposite sign from $\ce{CH}$ or $\ce{CH3}$ signals.