Personally I stick with apache2 backend but have CloudFlare front-end which is nginx. The only hosting I've seen that uses litespeed is cpanel. I prefer to configure things by hand making cpanel a no-go for me. I can't speak for its speed as I haven't tested it long enough to give you an accurate answer.
It can make your website faster if you have a poorly configured stack. The downside is that there are better free alternatives for this which cost nothing (unlike litespeed), there are a limited number of server systems this runs decent with, and sadly it costs more than it should for a license to run litespeed.
I have Lightspeed, nginx, and other server backends. They are all good, I have never really seen a speed increase, Lightspeed for me is more of a marketing. It is like Sentricon Termite Monitors. The Sentricon System is a hit or miss system, it is not the best but companies offer the service because of the name recognition.