For the connection yes it might be along with other variables mentioned by Amkam
For the difference
HR standard FR channel is broken down to two i.e the channel bit size remains the sampling rate is reduced. that is user gets every alternate slot on that specific TCH of that radio. for SAACH information of the second user the idle in the TCH multiframe is used to send the SAACH information of user B.
EFR vs FR there is difference in the number of speech and CRC bits, while using the same Full TCH channel . EFR gives better customer perceived voice quality due to greater CRC capability.
AMR works for both HR and FR and again gives a varity of codecs i.e CRC vs Speech bits combination mapped WRT BER in bad BER more CRC bits are used and in good BER less CRC bits are used.