The main difference between Swagger and HATEOAS IMO, which is not covered in the accepted answer, is, that Swagger is only needed for RPC’esque APIs. Such APIs, however, have actually hardly anything to do with REST.
There is a further, widespread misconception that anything exchanged via HTTP is automatically RESTful (~ in accordance with the REST archtitectural style), which it is not. REST just defines a set of constraints that are not choices or options but are mandatory. From start to finish. There is nothing wrong from being not RESTful, but it is wrong to term such an architecture REST.
Swagger describe the operations that can be performed on an endpoint and the payload (including headers and the expected representation formats) that needs to be sent to the service and also describe what a client might expect as response. This allows Swagger to be used both as documentation as well as testing-framework for the API. Due to the tight coupling of Swagger to the API it behaves much like a typical RPC service description, i.e. similar to WSDL files in SOAP or stub or skeletton classes in RMI or CORBA. If either the endpoint changes or something in the payload changes, clients implementing against a Swagger documentation will probably break over time just reintroducing the same problems typical RPC implementations have.
REST and HATEOAS, on the other side, are designed for disovery and further development. REST isn’t a protocol but an architectural style to start with that describes the interaction flow between a client and server in a distributed system. It basically took the concepts which made the Web so successful and translated it onto the application layer. So the same concepts that apply to the browsable Web also apply to REST. Therefore it is no miracle that also HATEOAS (the usage of and support for links, link relations and link names) behave similar to the Web.
On designing a REST architecture it is benefitial to think of a state machine where a server provides all of the information a client needs to take further actions. Asbjørn Ulsberg held a great talk back in 2016 where he explains affordances and how a state machine might be implemented through HATEOAS. Besides common or standardized media-types and relation names no out-of-band knowledge is necessary to interact with the service further. In the case of the toaster example Asbjørn gave in his talk, a toaster may have the states off, on, heating and idle where turning a toaster on will lead to a state transition from off to on followed by a transition to heating till a certain temperature is reached where the state is transitioned to idle and switches between idle and heating till the toaster is turned off.
HATOAS will provide a client with the information on the current state and include links a client can invoke to transition to the next state, i.e. turning the toaster off again. It’s important to stress here, that a client is provided by the server with every action the client might perform next. There is no need for a client implementor to consult any proprietary API documentation in order for a client to be able to interact with a REST service. Further, URIs do not have to be meaningful or designed to convey a semantical-expressive structure as clients will determine whether invoking that URI makes sense via the link-relation name. Such relation names are either specified by IANA, by a common approach such as Dublin Core or schema.org or by absolut URIs acting as extension attributes which might point to a human-readable description, which further might be propagated to the user via mouse-over tooltips or such.
I hope you can see by yourself that Swagger is only needed to describe RPC Web-APIs rather than applications that follow the REST architectural design. Messages exchanged via REST APIs should include all the information needed by a client to make informed choices on the next state transition. As such it is benefitial to design such message flows and interactions as state machine.
Update:
How are Swagger and HATEOAS mutually exclusive? The former documents your endpoints (making auto-generating code possible) and the latter adds meta-information to your endpoints which tell the consumer what they can do (i.e. which other endpoints are available). These are very different things.
I never stated that they are mutually exclusive, just that they serve two different purposes, where if you follow one approach the other gets more or less useless. Using both does not make any sense though.
Let’s move the discussion to the Web domain as this is probably more easily understandable and REST is de facto just a generalization of the concepts used on the Web, so doing this step is just natural and also a good recommendation in terms of designing REST architectures in general. Think of a case where you as a user want to send some data to the server. You have never used the service before so you basically don’t know how a request has to look like.
In Swagger you would call the endpoint documentation, select the option that most likely might solve your task, read up on how the request needs to look like and hack a test-case into your application that ends up generating a HTTP request that is sent to the respective location. Auto-generating code might spare you some hacking time, though you still need to integrate the stub classes into your application and test the whole thing at least once just to be safe. If you later on need to integrate a second service of that API or of yet an other API in general, you need to start from the beginning and look up the Swagger documentation, generate or hack the interaction code and integrate it into your domain. Plenty of manual steps involved and in cases of API changes you need to update your client as otherwise it might stop working.
In the Web example however, you just start your browser/Web client, invoke the respective URI that allows you to send the data to the server and the server will most likely send you a HTML form you just need to fill out and click the send button which automatically sends the request to the server which will start to process it. This is HATEOAS. You used the given controls to drive your workflow. The server taught your client every little detail it needed to make a valid request. It served your client with the target URI to send the request to, the HTTP method it should use and most often also implicitly the media type the payload should be in. In addition to that it also gave your clients a skeleton of the expected and/or supported elements the payload should contain. I.e. the form may require you to fill out a couple of input fields, select among a given set of choices or use some other controls such as a date or time picker value that is translated to a valid date or time representation for you. All you needed to do was to invoke the respective resource in your Web client. No auto-generation, no integration into your browser/application. Using other services (from the same or different providers) will, most likely, just work the same way so no need to change or update your HTTP client (browser) as long as the media-type request and responses are exchanged are supported.
In the case where you rely on Swagger RPC’esque documentation, that documentation is the truth on how to interact with the service. Mixing in some HATEOAS information doesn’t provide you any benefits. In the Swagger case, carrying around additional meta-information that bloat up the request/response for no obvious reasons, as all the required information is given in the reference documentation, will, with some certainty, lead to people starting questioning the sanity of the developers of that service and ask for payload reduction. Just look here at SO for a while and you will find enough question asking on how to optimize the interaction further and further and reducing message size to a minimum as they process every little request and don’t make use of response caching at all. In the HATEOAS case, pointing to an external reference is just useless as peers in such an architecture most likely already have support for the required necessities, such as URI, HTTP and the respective media types, implemented into it. In cases where custom media-types are used, support can be added at runtime via plug-ins or add-ons dynamically (if supported).
So, Swagger and HATEOAS are not mutually exclusive but the other gets more or less useless once you decided for one route or the other.