You need to provide more information to be able to select between the two. Choosing between the two usually is the product of preference from a development perspective and the various non functional requirements you will have.
Consider Azure Functions as a way to leverage native bindings to other Azure services (eg the Database), with a specific programming model. Typically Azure Functions are used in a Serverless Consumption billing mode, but can also be associated with a regular App Service Plan for predictable billing.
App Service is a much broader capability of application hosting and integration with other services is done via SDK.
You have many options for hosting your API in Azure, I'd suggest you consult this part of the docs https://learn.microsoft.com/en-us/azure/architecture/guide/technology-choices/compute-decision-tree