I can't just put the SSD into the new PC and boot it up?
No, you can't put the i-3 into the HP. Different hardware completely. Round peg into square hole.
No. You're buying machine X.
You can't expect to install a hard drive installation from machine Y into machine X and expect to live happily ever after.
2 issues:
1: the hardware is significantly different.
2: the Windows license on the SSD came from somewhere. I don't know where that somewhere is, but it surely isn't from machine X that you don't even own yet. The installation on the SSD is tied to whatever machine it was in when you installed Windows to the SSD--unless it's a retail license, whcih is doubtful.
You could download a legit Windows ISO, burn it to a disk, and install to the SSD after you move it to the new PC, wiping out whatever is now on the SSD. Then activate with the Product Key that will be on a sticker on your new PC.
or
Use imaging to transfer the Windows installation on the new PC's hard drive to the SSD.
But you can't simply install the SSD into the new PC and start up. It may run temporarily, but would fail activation at some point because of the new hardware----which Microsoft will become aware of.