Acid-free foamcore is similar to regular foam core, but acids and lignin in the paper surface have been removed so that acid-free foam core won’t harm your art.
If you have an inexpensive poster that can be easily replaced, you can save a little bit of money by going with the regular foam core. But if your art is original and irreplaceable, use acid-free foam core.
The difference between one-eighth and three-sixteenths is literally just the thickness. If your frame isn’t very deep, choose the one-eighth inch foam core. But if your frame is deep enough to accommodate an extra one-sixteenth inch thickness, choose the three-sixteenths foamcore, as it’s more rigid than one-eight inch foam core.