I suspect the hysteresis (difference between switch on and switch off points), of the old stat, is wider than the new one, plus you are perhaps setting it a bit higher in temperature.
A separate issue, might be the boiler stat setting. The boiler needs to output it's heated water, around 10C higher than the setting of the cylinder stat - otherwise the boiler will cycle on and off forever, trying to heat the cylinder, and trip it's stat to off. For instance - cylinder stat set at 55C, the boilers own stat, needs to be at least 65C, then tested to ensure the cylinder stat does turn off.
A separate issue, might be the boiler stat setting. The boiler needs to output it's heated water, around 10C higher than the setting of the cylinder stat - otherwise the boiler will cycle on and off forever, trying to heat the cylinder, and trip it's stat to off. For instance - cylinder stat set at 55C, the boilers own stat, needs to be at least 65C, then tested to ensure the cylinder stat does turn off.