Foxfi works great on the S3.... with any foxfi install, things are smoother once you reboot the phone.. then its set for good "unless you uninstall".
It connects my S3 to the internet much faster than using the stock hotspot on the phone.... "I have free hotspot on my phone through my share everything plan" , my computer identifies it almost instantly and gives me internet access, whereas the stock one takes about 30 seconds or so.
Its hard to test, but foxfi may be faster too.
Still not sure if its verizon and going around them is faster, or whether its the phone..... I"m thinking some samsung phones have some issues as hinted on foxfi's site.... something to do with conflicting with wifi being connected to a SSID when trying to activate hotspot.