There are major difference between epsom salt and sea salt. There are a lot of reasons that epsom salt isn't used for near as much as it use to be (i won't get into all of that, because I'd have to go into the chemistry of it and this post would end up taking half an hour to read).
Sea salt (sodium chloride) has a 98% purity level (the other 2% is trace minerals), it contains up to 81 all natural minerals (depending on where it is from) that are essential to the body, when mixed properly it creates a completely isotonic solution that is safe to use on fresh piercings (isotonic means it is the same ph level as the human body), sea salt has soothing properties that relieves irritation in pierings, the list goes on.
Epsom Salt (Magnesium Sulfate Heptahydrate) is for use in relieving muscle pain mostly. It is also used in some cosmetics such as exfoliants, it's also used in lawncare products, deodorants, and many other things that you would not want on a piercing. It doesn't even contain sodium which is what salt is composed of. If you read the box that it comes in, it even says not to use on skin infections or open wounds and a piercing is an open wound. Here is a link to an MSDS (Material Safety Data Sheet) for Epsom Salt:
http://www.sciencestuff.com/msds/C2030.html
Here is the link to the APP (Association of Professional Piercers) website for piercing aftercare: http://www.safepiercing.org
Needless to say, epsom salt on a piercing might have been what you were told to use 10 years ago, but it's not something you'll find a piercer recommending for use today. As with modern medicine, body piercing is a science and science is constantly evolving to new better ways of doing things.