If you turn on two faucets at the same time, like the kitchen and the bathroom, I bet it will run constantly.
All that is happening is the water pressure is reaching it's max pressure and turning off the pump. Then when water runs, pressure lowers, and the pump turns back on.
Unfortunately, it only take about a 1/4 cup of water for the cycling on-off-on to occur, as there is no give in the lines at all.
The pump is doing exactly what it was designed to do. Chances are, the threshold between on and off and on is much less than on your old pump. As DrewE stated above, and accumulator tank will cause the cycling between on and off to be longer.
Your pump is doing exactly what it's suppose to do. It's fine.
When running my shower, if only the hot water is on, the pump cycles on and off pretty fast. Once I get hot water flow, I turn on the cold to create warm water. As soon as the cold is turned on, the pump runs steady.
It runs steady when I flush the toilet. It pulsates at the kitchen sink, unless I turn both hot and cold on full.
Your pump is fine.