The motor should be the same but you may notice it more if the phase is lost at the service. When some of the lights go out or the A/C stops working or whatever.
If a branch OCPD fuse opens, that phase has no power on it. How would that be different from not having power from a loss of phase further up stream?
Rhetorical question. Here's how. If you lost a phase to a wide are in a facility, every 3phase motor that was already spinning in that facility is now acting as a rotary phase converter, trying to recreate that missing 3rd phase from the other two. So what you get is an incomplete voltage on the missing phase, as opposed to zero volts on the missing branch. So you get a severe voltage imbalance on every motor affected by the phase loss, but not a complete phase loss. That's why voltage based phase monitors are sometimes fooled, they see the phantom voltage from the spinning motors and fail to trip.
If you are talking about a small installation with only one 3phase motor, like a remote pumping plant or something, then there is no difference.
Assuming of course that the phase loss is NOT on one of the phases supplying control power, as meter nerd said.