AWS privatelink vs AWS Private API Gateway

PrivateLink is great for SaaS providers to give access to their application across VPC boundaries. It works for TCP connections so it's not restricted to providing API, web requests or anything - raw TCP is just fine.

API Gateway is an API service - it requires you to define the API and then share it with the "other" account. This is no different to PrivateLink in that sense except that you are offering a HTTPS-based API. It has the advantage over PrivateLink in that you have flexibility in how requests are filtered, routed and authenticated within the API Gateway service. API Gateway also has session timeout limits and data transfer limits that are lower than PrivateLink: https://docs.aws.amazon.com/apigateway/latest/developerguide/limits.html

So the answer is "it depends" - how is your application designed? What protocols does it use? What do you need the connectivity piece (API Gateway/PrivateLink) to do?

In terms of speed - I assume you mean data transfer speed? PrivateLink is likely to be a little quicker but only because the TCP session is between the client and the target via PrivateLink and a NLB. With API Gateway the TCP session is terminated in API Gateway and another session is then established to the "target" (not really the right word, hence the quotation marks) - whatever is processing the API Request. So there is more work happening which will make it slightly slower.