Typekit is now Adobe Fonts. You must have an Adobe subscription to use it, and the subscription holder is the one who holds the rights to the fonts.
As for which one to use...
If it is a straight web project, and you want the client to be responsible for all costs, stick with Google Fonts and limit your design to what’s available. Google’s library is limited to fonts which are licensed freely.
Adobe Fonts is not just web fonts. It is fonts to use in all Adobe apps and for both print and digital collateral. So if you are doing branding work for a client, for example, and you have a specific font that you are using across different collateral, you would naturally want to use this font on the web. The Adobe Fonts library offers many commercial licensed fonts that are not available in Google Fonts.