Both lenses have the same focal length range and are very close in terms of the maximum aperture at any particular focal length. The main differences are:
- One has Vibration Reduction (VR) also known as image stabilization and one doesn't.
- The way that each one autofocuses (or doesn't autofocus) with specific camera bodies.
The AF-P DX Nikkor 70-300mm f/4.5-6.3G ED VR is a newer design that will work on your D5500. As the model name indicates, in includes VR. It also has a new type of stepping autofocus motor in the lens. Nikon designates all lenses with this type of AF motor as AF-P lenses.
Although your D5500 can control the AF motor in this and any other AF-P lens, many other slightly older Nikon DSLRs, including most all of the very high end ones, cannot. So if you ever consider buying a used upper tier Nikon DSLR, you'll need to check and see if that body is compatible with Nikon AF-P lenses. The only models currently capable of autofocusing or manually focusing the relatively new AF-P lenses are the D500, D3300, D3400, D5300, D5500, Df, D5200, D7100, D7200, D750, D810, . Most of these models require the most recent firmware version to be installed to be compatible with AF-P lenses. One could also assume that as new models are introduced in the D3x00, D5x00, and D7x00 lines they will be AF-P compatible.
Only the D500, D3300, D3400, D5500, D5600, and D7500 are fully compatible and can control whether the lens' VR is active or turned off, since there is no switch on the lens itself.
When using AF-P lenses that have VR with the D5200, D5300, D7100, D7200, D750, D810, Df, and D5, VR will be active and can not be turned off.
From Thom Hogan's review of the lens (bold type added):
The current list of full compatibility is D500, D3300, D3400, D5300, and D5500 (but make sure that you’re using the latest firmware). The following cameras have no way to set VR and may change focus position if the camera’s meter goes inactive (requiring you to re-obtain focus): Df, D5200, D7100, D7200, D750, and D810. All other Nikon DSLRs should be considered completely incompatible with this lens. Further note that this is a fly-by-wire lens, so while it may appear to mount on those other Nikon DSLRs, you can’t focus the lens manually, thus the incompatibility warning.
The AF Zoom-Nikkor 70-300mm f/4-5.6G BK is an older lens design that does not have an autofocus motor built into the lens at all. It does have a connection for the screw drive AF motor on higher end Nikon bodies:The older D70/D70s/D80/D90/D100/D200/D300/D300s cameras as well as the newer D500/D600/D610/D700/D750/D800/D810, and the professional D1/D2/D3/D4/D5 series of bodies. The D3x00 and D5x00 series, which includes your D5500, does not have an AF motor built into the camera body. This lens would need to be manually focused when using it with a D5500. Without the manual focusing aids found on most older manual focus SLRs, this would be a challenge.
Here's a link to a chart of the different types of Nikon lenses (AI, AF, AF-S, AF-P, E-type) and which bodies are fully compatible, have limited compatibility, or are not compatible with each type.