Okay if you haven't read chapter 621 go away and read it.
So this last arc of Naruto has introduced 2 new Sage styles we never knew about but I thought it would be fun to discuss the differences and origins of each. We know Sage mode involves absorbing natural energy from ones surrounding and combining it with ones physical & spiritual energy to enter the sage state.
I think we have to start at the beginning and figure out that Natural energy may in essence be similar to chakra the way it can be channeled into different types like Fire, Water, Earth, Lighting & Wind. You can also extend that to Yin & Yang properties. Each style of Sage mode focuses on a different way of channeling natural energy and produces different results yet shares some similarities. They all boost the users basic stats in all areas and sense of perception increases but they also give greater advantages in some areas.
It was mentioned there are 3 places that specialize in the Sage art which are Mount Myōboku, Ryūchi Cave & Shikkotsu Forest.
The Frog styled Sage mode seems to focus on boosting physical stats so would probably be Yang heavy and most suited to direct combat. One gains incredible psychical strength as we saw tiny frogs lift huge statues that probably weighed tons. Naruto was able to extend his reach with this sage mode as well without physical contact.
Frog Kata's training takes advantage of the increased strength and agility of this mode.
The Snake styled Sage mode seems to focus on physical alteration and perception. This seems to scream out mental energy and Yin styled which controls imagination. Look what happened to Juugo's clan they literally lost mental control from absorbing this style of energy which seems similar to Uchiha clans curse due to high concentrations of mental Yin styled energy.
The new style probably just revealed in chapter 621 that Hashirama Senju uses which seems to focus on life giving energy that effects growth and healing. This may very well connect with the slugs Tsunade summons but that is just a theory. This style of senjutsu is for healing arts or in Hashrima's case since he uses his chakra to turn into a source of life for growth in his life giving jutsu that enhances and feeds his living Mokutan jutsu.
The 4th Style
I have an idea that there is fourth style of Sage art used only by those who can master all styles like the Sage of the 6 paths. It could be considered some kind of divine Sage mode that is the origin of the other 3 as it was far too difficult for anyone to master.
There maybe different variations of the main sage arts but they all generally utilize similar aspects as the 3 above.