LQAS is not a sample design or a sampling methods. It is an approach to data analysis. The confusion occurs because these sorts of methods can be applied to data as it is collected so that sampling can stop.
Typically, an LQAS study will be quicker and cheaper than another approach but the results will usually be limited to making classifications (e.g. Is the prevalence of drug resistance above or below a certain level?) rather than estimates (e.g. The prevalence of drug resistance is X% with a 95% CI of Y% - Z%) as is usually provided by other approaches.
Comparison is not easy because the aims are different. I find LQAS very useful for quality control applications. In our field this can be about coverage or other delivery standards or when costs need to be kept low. LQAS-like approaches are often very much quicker and cheaper than other methods. If you find that you often make estimates and than apply a classification (e.g. prevalence is high) then you may want to consider saving time and money with LQAS.
Most critics of LQAS-like methods mistake it for something that it is not. There is little value in critiques that say LQAS is rubbish because it only classifies when the method is designed to only classify. This is a bit like criticising cats because they do not bark and do not deter burglars when the reason for having a cat is to control disease vectors such as mice and rats.
One comparison that can be made is to use both methods to make a classification and compare the classifications. Do not use a single small LQAS sample for estimates as any comparison will lack power due the small sample size in the LQAS arm.
I hope this is of some use.