Clarifications:
- REST is an architectural style (repeated for completeness, you seem to be clear on this point).
- The World Wide Web is a reference application, which (mostly) demonstrates that architectural style.
- HTTP is one of three "core" technologies in the web stack.
I don't find a clear difference between HTTP and REST then if REST is implemented based on HTTP.
The short answer is that HTTP alone is not sufficient.
Jim Webber (2011)
HTTP is an application protocol whose application domain is the transfer of documents across a network.
We had applications to transfer documents across a network prior to HTTP -- file transfer protocol, gopher, wais... all of which had negligible penetration into the general public. In contrast, the web was catastrophically successful; it was the killer app of the internet.
Fielding's thesis -- in particular Chapter 6: Experience and Evaluation -- is, among other things, an exploration of the question "why was the web so successful?" with emphasis on the architectural constraints that protected the induced properties of the web.
In 2008, Leonard Richardson introduced his maturity heuristic for evaluating web services.
[URI + HTTP + HTML] form a technology stack for web services. When people design a web service they tend to pick some technologies from the bottom of the stack. You can judge them crudely by seeing whether they pick zero, one, two, or three technologies.
When I say you pick from the stack I don't mean that you'll ever find a web service that doesn't use HTTP at all or that has no URIs. I mean there's a class of web services that doesn't really get URIs or doesn't really get HTTP. If your REST radar is finely tuned you sense there's something wrong with these services, and you can talk about violation of the RESTful constraints, but it's kind of a bloodless way to talk. It's not clear why anyone should care.
These technologies don't implement poka-yoke. Which is to say, the technologies don't force you to use them completely, correctly, or to best effect.
Ian S Robinson covered similar ground in his 2010 talk The Counterintuitive Web.
In the talk I described how we can implement rich and interesting business processes in (RESTful) Web applications, but only if we think in terms of protocol resources, not coarse-grained domain resources. By embracing the Web as first and foremost a web of data, an open set of resource representations manipulated in the same-old-same-old ways using a closed set of verbs, our designs capture the behaviours most CRUD-based, data-centric applications so sorely lack.
This slide, taken from that talk, illustrates three different resource designs:

You can do all of these on HTTP, the protocol standard does not constrain you.
The design at the top of the slide has an RPC character: the business protocol is executed by sending many different messages to a single endpoint. Participation in the business protocol is limited to those components in the conversation that recognize this particular interface; in short, you can't achieve scale with out-of-the-box standard http components.
The design at the bottom has a REST character: many endpoints (resources), but the business protocol is executed by via a constrained set of messages (ie, the uniform interface) that have well specified semantics. By moving the protocol complexity from the messages to the resources, message exchange becomes business protocol agnostic -- you can achieve scale with standard components because they can participate in the exchange of representations without needing any specialization at all.
The one in the middle is Rails -- Jim Webber, 2011.
The notion of the uniform interface is critical in the success of the web; it's the constraint that allows clients (browsers, crawlers) and intermediaries (caches, reverse proxies) to develop independently of servers.
What features/constraints can be achieved with REST based API that cannot be with only HTTP based API
Fielding's thesis gives us this definition of the uniform interface:
REST is defined by four interface constraints: identification of resources; manipulation of resources through representations; self-descriptive messages; and, hypermedia as the engine of application state.
Hypermedia is huge, but it's not HTTP -- in the web stack, hypermedia support comes from HTML. It's the highest level in Richardson's model; the technology that is most often misunderstood when implementing a web service.
As Fielding (2008) bluntly clarified, this architectural constraint is *not optional:
What needs to be done to make the REST architectural style clear on the notion that hypertext is a constraint? In other words, if the engine of application state (and hence the API) is not being driven by hypertext, then it cannot be RESTful and cannot be a REST API. Period.
A REST API should be entered with no prior knowledge beyond the initial URI (bookmark) and set of standardized media types.... From that point on, all application state transitions must be driven by client selection of server-provided choices that are present in the received representations or implied by the user’s manipulation of those representations.
Fundamentally, consuming a REST API is like browsing wikipedia

Specialization and innovation depend on an open set. Note the implication: the open set is potentially changing and boundless. Change is a very real thing -- for reference, see just about any discussion of API versioning. Coupling independently developed clients to an open set of resources is completely unreasonable.
But with hypertext, can use a closed set of resources (bookmarks) to provide representations to the client that direct them to today's open set of resources, and then change the bookmarked representations tomorrow when you innovate.
It's a lot of work though -- much easier in the short term to communicate the available resources to the client out of band (ie, API documentation) which allows you to use representations that don't specify hypermedia control elements (example: application/json).
REST is intended for long-lived network-based applications that span multiple organizations. If you don’t see a need for the constraints, then don’t use them.
To give "long lived" a sense of scale -- httpd was first implemented over 25 years ago, and the cutting edge of HTTP is version 2. HTML is more volatile (and somewhat older); it is all the way up to version 5 (numbering somewhat confused by the fact that WHATWG considers HTML to be a living standard).
There are some API like Jersey JAX-RS which provide API implementation for REST based on HTTP. What extra features they add on HTTP basics?
Umm... not a lot?
That's not a particularly charitable answer, and Fielding was one of the expert members, so you are invited to take my skepticism with a grain of salt.
But here's what Marc Hadley had to say in 2008
I think the API encourages a resource-centric view and makes developers think about the identifiers of their resources and the methods they support. Declarative support for content negotiation works well and the default resource life-cycle encourages a stateless approach.
Suggestion, keep "think about the identifiers of their resources" in mind as you watch Stefan Tilkov's talk REST: I don't think it means what you think it does (slides).
Here's Marc Hadley on the weaknesses of the work
If I had to identify a weakness it would have to be limited support for hypermedia as the engine of state - whilst we provide good support for extracting information from request URIs and building URIs to resources, its still very much left to the developer to use hypermedia in representations appropriately.
Yes, generation and parsing of nicely designed URI has value. However, nicely designed identifiers are not among the REST architectural constraints. Hypermedia is.
In conclusion, if you are looking to understand the distinction between HTTP and REST, JAX-RS won't help.