Fill the gap with foam backer rod the use Sonneborn SL1 sealant. It is "self leveling" so it will fill all the voids and look great. It will run down hill fast so you should only use it on the horizontal joints. Use the matching NP-1 for the slope joints. You can get it at Maxwell's or SSI on Lincoln & 36th. You should stay away from silicone for concrete joints.
Why is this?
Also, if the joints run all the way to the edge of the drive, should you cut the foam backer a bit back from the edge and fill the entire edge with sealant? I mean, if you run foam to the edge and only seal on top of it, then water can run under the sealant and along the foam, defeating the purpose of sealing, right?