TLDR: You got the wrong replacement burner. Take it back and get a 3-wire one, but read to the end to check that it's actually the burner at fault and not the thermostat.
New update: after reading the service manual you linked, I see the old burner is not an old-style 3-level low/medium/high, but the TLDR above stays the same.
Details:
You old 3-wire burner has 2 heating elements, controlled by 2 separate temperature-controlled switches in the thermostat (control knob). It's described on page-12 of the service manual.
That 4th "white wire" running to the center of the burner is not a wire - it's a capillary tube which allows the thermostat to sense the temperature of the surface of the burner.
So depending on the current surface temperature and the position of the control knob, the thermostat switches one or both of the heating elements on or off to try to maintain a reasonably constant temperature.
Your new burner is a single element with, but it's intended to be controlled by a 'simmerstat', which is a switch which continuously turns the heating element on & off, and achieves a variable power level by adjusting the ratio of on-time to off-time. It's described on page-11 of he service manual.
This type of control does not have a thermostat like your-old 3-wire burner, and does not maintain a constant surface temperature.
Now for the problem. The old burner's thermostat control using that capillary tube is not something you can disconnect & reconnect like a wire - it's a sealed unit from the thermostat, through the capillary tube, to the temperature sensor in the center of the burner.
It may be possible to remove the temperature sensor from the center of the burner (it looks like it's held in by some spring clips), but there's nowhere to fit it on your new burner.
Your old thermostat control knob will not work if that temperature sensor it not fitted correctly - in all likelihood no matter what setting you turn it to, it will always be on its maximum heat output because the thermostat will never know that it's hot.
In order to make use of the new 2-wire burner you got you'll need to replace the existing thermostat control knob with a matching simmerstat control.
The better idea would still be to send back the 2-wire burner and figure out what the correct 3-wire burner's part number is - but before you order a new one check that it's actually the burner that's failed and not the thermostat!
Measure the resistance across the 2-1 and 2-4 terminals of the old burner to check if it reads nominally 45 & 40-ohms as indicated in the manual.
If those readings are correct, then it's not the burner that's at fault and what you probably need is a new thermostat instead.
If you're comfortable testing live wiring, then disconnect the old burner, and make sure the bare wires are far away from each other and from any other metal surfaces. Turn the burner on to a medium/high level, and measure the voltage between the 2-1 and 2-4 wires. You should measure 240V in each case. If you do not see the correct measurements then it's the thermostat at fault.
