Spring abstracts you from the very very very large list of http status code. That is the idea of the exceptions. Take a look into org.springframework.web.client.RestClientException hierarchy:
You have a bunch of classes to map the most common situations when dealing with http responses. The http codes list is really large, you won't want write code to handle each situation. But for example, take a look into the HttpClientErrorException sub-hierarchy. You have a single exception to map any 4xx kind of error. If you need to go deep, then you can. But with just catching HttpClientErrorException, you can handle any situation where bad data was provided to the service.
The DefaultResponseErrorHandler is really simple and solid. If the response status code is not from the family of 2xx, it just returns true for the hasError method.