First, the 339 has an open collector output. This means that it can't output a voltage, only pull down a voltage provided by a pull up resistor or similar. The 339 flips the output based on which input has the higher voltage. It doesn't act as an amplifier. If you want to amplify the voltage over the sense resistor you need an op amp. Also, the op amp must be able to handle signals close to the power supply. In your application the signals are within 1V of the negative power supply of the op amp, ground in this case. Not all op amps can do this.