Water softener and RO filter
You are not wrong in your understanding of a water softener, nor are you wrong in your understanding of a reverse osmosis filtration system.


  Does the RO filter care whether the supply water is salty or hard?


I would like to make the point that your RO filter is incapable of caring about the hardness of the water. This is more about carefully choosing your words more than anything else. I was also tripped up by this when I first began chemistry, but it was something I learned to do and I challenge you to do the same.


  Is it more effective at filtering one or the other?


Yes, but only slightly. Here you can find a list of the typical rejection characteristics of RO filters. The lower bound for the percent amount of multivalent ions ($\ce{Ca^{2+}}$ and $\ce{Mg^{2+}}$) filtered is higher than the lower bound for the percent amount of monovalent ions ($\ce{Na+}$ and $\ce{K+}$) filtered, but the upper bounds are very close in value. 

Overall you could say that RO systems are more efficient at filtering hard water, but the difference is not very large. Most systems will likely be performing considerably above the lower bounds of the typical rejection characteristics anyway, so this should not be a deciding factor in deciding to feed your RO hard or soft water.


  Is one going to influence the life of the RO filter more than the other?


Yes! In an ideal RO filtration system, all of the filtered materials would simply drain away, and pure water would flow to the storage tank, but nothing is ideal. If you feed hard water into your RO system, there will be scaling on the semi-permeable membrane that will eventually reduce its filtered water output. This can cause lack of flow will create a potentially damaging buildup of pressure elsewhere in the system. I would therefore recommend that you keep your setup the way it is!