Where ICO came from
ICO is the Windows icon container, dating to Windows 1.0 in 1985. Its distinguishing feature is that one file holds several resolutions at once, so the system picks the sharpest version for the context. That is exactly what browsers want from a favicon, which is why a 1985 Microsoft format is still sitting at /favicon.ico on most of the web.
The technical picture
Compression: Bundles PNG or BMP data per size. Transparency: Full alpha in the PNG-based variants. Animation: Not supported. Colour: 8-bit per channel. Typical file size relative to WebP: small by definition — icons are tiny.
Where it opens
Browsers and Windows. This is usually the deciding factor when choosing between ICO and WebP — WebP renders in every browser but is uneven in desktop software, while ICO has a very different support profile.
Converting ICO to WebP
Converting an ICO to WebP pulls the largest bundled resolution out into a normal image you can use in a page, a README or a design file. Start from the largest, squarest source you have. Icons are generated by downscaling, and a busy or non-square image turns to mush at 16x16.
Converting WebP to ICO
The usual job is turning a logo you have as a .webp into a site favicon. The output bundles 16x16, 32x32 and 48x48 in one file so browsers and Windows shortcuts each pick the size they need. Expect the ICO output to be small by definition — icons are tiny — that difference is the price of getting a format the destination will actually accept.