The major benefit is that it's possible to synthesize methane from carbon dioxide and hydrogen. Which is way easier let's say on Mars, than obtaining kerosene.
Wait, I'm confused... LNG is pure methane. Are there rockets that use methane that aren't storing it as a liquid? I would have to guess that any rocket using methane would pressurize it enough to be in liquid form for storage, anyway. Surely they wouldn't use methane in a gas state... It would take up far too much space to be practical.
LNG is 85-95% methane, typically, with ethane making up most of the rest. It contains small amounts of propane and butane.
I hadn't known that SpaceX was using a special grade of ~100% liquefied CH4 as nickik says above.
EDIT: shale gas is richer in ethane and higher hydrocarbons than traditional sources of natural gas. Maybe SpaceX is sticking to 100% methane just so they don't have to ponder variability of composition depending on how gas is being extracted.
Methane can be synthesized from CO2 and hydrogen. More complex hydrocarbons are way more difficult to produce. So pure methane based engine is a better method for using on other planets.
When you liquefy natural gas, ethane and methane liquefy at different times, so it's relatively easy to purify methane. I don't know the price difference, do you?
See:
https://en.wikipedia.org/wiki/Sabatier_reaction
http://www.co2club.it/agenda/abstracts/19_Barbarossa.pdf