Guy Gardener is on the right track. The basic tenet of troubleshooting is "isolate the variables." In other words, suspect everything and test each factor one at a time. Guy suggested checking the power supply—obviously you got a new one when you bought the new drive. Or is this a tiny drive meant to be powered by the data port? I've run into many "portable" USB drives that need more juice than a single USB port can supply, and not all USB ports are equal. Some can push more wattage than others. (This applies to other powered ports, too, but USB is the one I generally find being anemic.)
Is the data cable good? If the drive and the computer don't see each other, the drive won't spin up.
Is the port on the computer bad? You can check that by trying the drive on another computer and/or connecting a different drive (or other device) to the computer.
And so on. Check each possible point of failure one at a time.