I agree, in general, with the conclusions arrived at and listed here by Doc Snow, pinkboxer and TheJman. However, in my opinion, "Watch your tongue" should actually be "Hold your tongue". I think this sharpens the difference between the two phrase's meanings. "Hold (or watch) your tongue" simply means that you really shouldn't say anything at all, (IE: Keep your trap shut!), whereas "Watch your mouth" strongly implies that you are not to say anything inappropriate, due to being in 'mixed company'.
'Mixed company' is commonly a group of men and women collectively. It usually is a circumstance where saying the wrong thing could lead to embarrassment. Otherwise, it may mean a group of people with differing national or geographic backgrounds, opposing political views, different cultures, religions, life styles and even a mix of races. It is where something said, without caution, might be taken negatively, resulting in arguments, physical fights &/or out-right war, thereby leading to the possibility of causing serious and long-term ill-feelings and sometimes even death. It all depends on the situation; It could be anything from a 'blue' or dirty (filthy? sexist?) joke, to a (deliberate) nasty slur.
>>> I hope this helps toward clarifying the respective differences in similar terminology, both in meaning and usage. Therefore, the basic answer to the initial question is No, although some might say "Yes, sort of, but not exactly."