If the VFD is bypassed, the motor speed is no longer associated with the VFD and is reverted back to S = F x n/120 where S = Synchronous speed F = frequency, n = number of motor poles, and 120 is a mathematical constant.
If the motor runs at a lower speed in VFD mode based on a 60Hz output, that is indicative of a problem either in the sizing of the VFD or in the programming of it. So for example if you have the VFD under sized for the load, when the VFD attempts to get the load to 60hz, the current increases to a value higher than what you have programmed the VFD to allow, and/or the VFD is capable of. in addition someone likely enabled a feature in the VFD, that only VFDs are capable of providing, which limits that current going to the motor. But HOW it accomplishes that is by ignoring the speed command you WANT to run it at, and instead runs the motor at the highest speed it CAN run it, given the limitation on current going to the motor. Bottom line, it might be doing EXACTLY what it is supposed to be doing.
But if, on the other other hand, you are assuming the motor is running slower based on the current being lower when on the VFD, that might simply be measurement error. It's difficult to measure the output current of a VFD with a simple clamp-on meter, and if you are measuring current on the INPUT of the VFD, that's a whole different kettle of fish involving power factor that's too long to go in to needlessly. But if you ARE measuring current on the input side, let us know and I (or someone else) will explain to you what you are seeing.