Working with current feedback opamps [CFA]
There is nothing is wrong with your circuit. The wide pulse width is because you're feeding your diff_amp_out to a comparator.

It can only supply you with a pulse of full voltage if the input (in this case) is even slightly above zero volts. I like how your scope shows the tiny delays from stage to stage.

You could adjust the 'trip' level of the comparator, but the output will always be +5&nbsp;V.

If you just want to amplify diff_amp_out, add another AD8003 op-amp and set your gain to fit your needs. At least you keep your true signal pulse width. Admittedly, working with these nS pulses is tough.

CFA amps are designed for current feed back, which lowers the input impedance, but greatly increases its bandwidth and high frequency response compared to VFA.

All video and RF amplifiers are the CFA type. The VFAs are best for audio and precision DC measurements.