Well, I haven't been to LSE so i can't compare but academics wise I would say St Andrews is average, it was lower than I expected however my expectations were high considering St Andrews rankings (at the time St Andrews was ranked 4th/6th so I kind of expected a strongly ambitious, nerdy bunch studying at the library often).
I feel like career wise, people in St Andrews don't seem as ambitious. I say this because I just don't see much news about St Andrews students winning competitions, participating in global events, winning major scholarships etc. I don't hear much about major companies coming to St Andrews to recruit either. This may be in part due to the type of student who comes to St Andrews but it doesn't have the cutthroat competition that I came to expect. This is a good thing in my opinion, but it depends on what you are looking for.
I don't know about LSE, but for my experience going to an ultra-competitive high school in a major city, I feel like LSE will be more cutthroat as its definitely going to attract more ambitious types. St Andrews the town also seems to lead people to a more relaxing time (i've certainly become much more study focused and have done fewer extracurriculars), it is a small town far from London, you will make friends and have a good time, but if you have big plans for your career I would recommend LSE.
Hope this helps.