1) Securing your back-end APIs - If you have a legacy back-end web service that has a basic authentication scheme, you can add some additional security by placing APIM in front, and requiring subscription keys. Leverage your existing firewall to ensure only your APIM instance can communicate with your back-end API, and you've basically added a layer of protection.
2) Lift and shift - there are always going to be clients that don't want to update their clients to use a newer API; in some cases you can make a newer API look like an older one by implementing some complex policies in APIM. You can also do the opposite, making older APIs look new, such as making an XML back-end accept both JSON and XML.
3) Centralizing your APIs - if you've acquired another company and want to make their API set look as if it's a part of the larger whole, APIM is an easy way to provide a consistent front-end interface for developers.
To my opinion Swagger Open Source is a very good tool to quickly produce an api documentation, that how I use it.Swagger can also produce an open api file to generate an api for our application server (Inersystems Iris) but we don't use this functionality for now. We plan to use more Swagger features in the future.
Lack of robustness is a bit of an issue. Several other providers offer more options and capabilities, but then, they are lacking in interface ease.
As with anything Azure, pricing is really hard to stay on top of. I always find that you really don’t know what you’re paying for until you get the bill. Having an excellent Azure Administrator can help resolve that.
Integrating with app services outside of Azure can be a challenge, or at least much more challenging than just using Azure App Services.
We use swagger only to generate the API documentation, for testing the APIs, we use Postman. We don't find an other tools for producing API documentation in web pages and also in pdf files. Postman is a good complement for testing APIs on InterSystems IRIS (a powerful application server and DB manager) which implements these APIs.