Just reading the docs, they seem very similar to me so I can't really discern why to use one over the other. Although identity token seems better since it has custom attributes on it from the user pool (eg: custom:blah and the default ones like name and email).
Right now, I am working with an app that passes the access token back down to the browser so it can use it for making ajax REST calls (there is an auth filter that expects this access token and validates it). Could I just switch out the access token with the id token? The current validation logic is to just get the sub field (the uuid) from the access token, but this sub field is also present in the identity token (as well as practically every other attribute except the aud which I don't need). I just want to make sure I am understanding this right as it is confusing to me why both tokens exist and seem so similar.
The id_token is for your application to process, so you can get all the personal details for your user, like their name, age, email address etc. Generally speaking you shouldn't send this token anywhere else as it contains sensitive user data.
The access_token is used to call other 'external' services (and by external I include other AWS services - these are often called over http). It provides service access authorisation for your user without having to include their personal details.
On the face of it this appears slightly confusing as you can actually use the id_token to access services in the same way as the access_token. However, good practise is to use the access_token in this circumstance and if backend services need user data, they should look it up themselves in Cognito.
EDIT: If you need to authenticate an api call based on claims in the identity token, there are circumstances when this is perfectly valid. But be aware of what details are in the identity token, and whether those claims are suitable to send to the particular API. If you don't need to use any claims from the id_token, use the access_token as this reduces the amount of potentially sensitive data you are sending.