2 Answers

You said "Restful Services", That means you do have https/http endpoints at present & You would like to do API Management on top of same.  Absolutely, You can do same using Apigee Edge & You can integrate Apigee with Restful Services. 

Coming to your dilemma "it seems unusual to use different product for API management and ESB",

API Management & ESB are totally different concepts.  It's perfectly fine to use a different platform to manage same. 

Let me help you understand the difference, ESB is point to point.  API Management is creating an open ecosystem of APIs through API Products. 

You will think inside out when it comes to ESB but Apigee - API Management is all about thinking outside - in to solve the consumption demands. 

Many Apigee Customers use Apigee on top of existing ESBs.  It's perfectly fine to do so. 

More than anything we would like to hear your business use cases & Why you are planning to use API Management Platform ??  It's all about solving business problems using API Management Platform rather than doing point to point traditional integrations. 

What's your business use case to do API Management ? 

Feel free to get in touch with me using Refer button on right side bar if you are not comfortable sharing your business use cases in public community. 

I have worked on IBM API Connect and see it as an enhanced version of IBM DataPower Gateway with additional features/concepts like Developer Portal, API Products & Key Management. 

I have seen a usecase like,

If your existing services are developed using IBM DataPower then, it is preferred to use API Connect, because it is easy to integrate those services with Datapower policies in API Connect. 

Other than this we need to consider many other points before zeroing down on an API Management Product as it the entry point for your services. 

Apigee internally has a nice comparison doc which highlights Apigee benefits over other API Management Products.  If your company is in partnership with APigee.  Ask your sales rep for the doc to understand various other points. 

Speaking about API Management & ESB, I agree with @Anil Sagar point of view.  Within my organization, I have seen many accounts follow the same approach, where API Gateway is placed in front of ESB.  Where ESB is used for handling complex orchestration/routing & connecting two different machines etc.,