Is REST a better approach to doing Web Services or is SOAP? Or are they different tools for different problems? Or is it a nuanced issue - that is, is one slightly better in certain arenas than another, etc?
Bounty-Edit:
Now, almost three years later I would like to ask this question again - offering a bounty to encourage an indepth answer. I would especially appreciate information about those concepts and their relation to the PHP-universe and also modern high-end web-applications.
REST is an architecture invented by Roy Fielding and described in his dissertation Architectural Styles and the Design of Network-based Software Architectures. Roy is also the main author of HTTP - the protocol that defines document transfer over the World Wide Web. HTTP is a RESTful protocol. When developers talk about "using REST Web services" it is probably more accurate to say "using HTTP."
SOAP is a XML-based protocol that tunnels inside an HTTP request/response, so even if you use SOAP, you are using REST too. There is some debate over whether SOAP adds any significant functionality to basic HTTP.
Before authoring a Web service, I would recommend studying HTTP. Odds are your requirements can be implemented with functionality already defined in the spec, so other protocols won't be needed.
One quick point - transmission protocol and orchestration;
I use SOAP over TCP for speed, reliability and security reasons, including orchestrated machine to machine services (ESB) and to external services. Change the service definition, the orchestration raises an error from the WSDL change and its immediately obvious and can be rebuilt/deployed.
Not sure you can do the same with REST - I await being corrected or course! With REST, change the service definition - nothing knows about it until it returns 400 (or whatever).
i create a benchmark for find which of them are faster! i see this result:
for 1000 requests :
for 10,000 requests :
for 1,000,000 requests :
REST is a fundamentally different paradigm from SOAP. A good read on REST can be found here: How I explained REST to my wife.
If you don't have time to read it, here's the short version: REST is a bit of a paradigm shift by focusing on "nouns", and restraining the number of "verbs" you can apply to those nouns. The only allowed verbs are "get", "put", "post" and "delete". This differs from SOAP where many different verbs can be applied to many different nouns (i.e. many different functions).
For REST, the four verbs map to the corresponding HTTP requests, while the nouns are identified by URLs. This makes state management much more transparent than in SOAP, where its often unclear what state is on the server and what is on the client.
In practice though most of this falls away, and REST usually just refers to simple HTTP requests that return results in JSON, while SOAP is a more complex API that communicates by passing XML around. Both have their advantages and disadvantages, but I've found that in my experience REST is usually the better choice because you rarely if ever need the full functionality you get from SOAP.
I think that both has its own place. In my opinion:
SOAP: A better choice for integration between legacy/critical systems and a web/web-service system, on the foundation layer, where WS-* make sense (security, policy, etc.).
RESTful: A better choice for integration between websites, with public API, on the TOP of layer (VIEW, ie, javascripts taking calls to URIs).
Answering the 2012 refreshed (by the second bounty) question, and reviewing the today's results (other answers).
SOAP, pros and cons
About SOAP 1.2, advantages and drawbacks when comparing with "REST"... Well, since 2007 you can describe REST Web services with WSDL, and using SOAP protocol... That is, if you work a little harder, all W3C standards of the web services protocol stack can be REST!
It is a good starting point, because we can imagine a scenario in which all the philosophical and methodological discussions are temporarily avoided. We can compare technically "SOAP-REST" with "NON-SOAP-REST" in similar services,
SOAP-REST (="REST-SOAP"): as showed by L.Mandel, WSDL2 can describe a REST webservice, and, if we suppose that exemplified XML can be enveloped in SOAP, all the implementation will be "SOAP-REST".
NON-SOAP-REST: any REST web service that can not be SOAP... That is, "90%" of the well-knowed REST examples. Some not use XML (ex. typical AJAX RESTs use JSON instead), some use another XML strucutures, without the SOAP headers or rules. PS: to avoid informality, we can suppose REST level 2 in the comparisons.
Of course, to compare more conceptually, compare "NON-REST-SOAP" with "NON-SOAP-REST", as different modeling approaches. So, completing this taxonomy of web services:
NON-REST-SOAP: any SOAP web service that can not be REST... That is, "90%" of the well-knowed SOAP examples.
NON-REST-NEITHER-SOAP: yes, the universe of "web services modeling" comprises other things (ex. XML-RPC).
SOAP in the REST condictions
Comparing comparable things: SOAP-REST with NON-SOAP-REST.
PROS
Explaining some terms,
Contractual stability: for all kinds of contracts (as "written agreements"),
By the use of standars: all levels of the W3C stack are mutually compliant. REST, by other hand, is not a W3C or ISO standard, and have no normatized details about service's peripherals. So, as I, @DaveWoldrich(20 votes), @cynicalman(5), @Exitos(0) said before, in a context where are NEED FOR STANDARDS, you need SOAP.
By the use of best practices: the "verbose aspect" of the W3C stack implementations, translates relevant human/legal/juridic agreements.
Robustness: the safety of SOAP structure and headers. With metada communication (with the full expressiveness of XML) and verification you have an "insurance policy" against any changes or noise.
SOAP have "transactional reliability (...) deal with communication failures. SOAP has more controls around retry logic and thus can provide more end-to-end reliability and service guarantees", E. Terman.
Sorting pros by popularity,
Better tools (~70 votes): SOAP currently has the advantage of better tools, since 2007 and still 2012, because it is a well-defined and widely accepted standard. See @MarkCidade(27 votes), @DaveWoldrich(20), @JoshM(13), @TravisHeseman(9).
Standars compliance (25 votes): as I, @DaveWoldrich(20 votes), @cynicalman(5), @Exitos(0) said before, in a context where are NEED FOR STANDARDS, you need SOAP.
Robustness: insurance of SOAP headers, @JohnSaunders (8 votes).
CONS
SOAP strucuture is more complex (more than 300 votes): all answers here, and sources about "SOAP vs REST", manifest some degree of dislike with SOAP's redundancy and complexity. This is a natural consequence of the requirements for formal verification (see below), and for robustness (see above). "REST NON-SOAP" (and XML-RPC, the SOAP originator) can be more simple and informal.
The "only XML" restriction is a performance obstacle when using tiny services (~50 votes): see json.org/xml and this question, or this other one. This point is showed by @toluju(41), and others.
PS: as JSON is not a IETF standard, but we can consider a de facto standard for web software community.
Modeling services with SOAP
Now, we can add SOAP-NON-REST with NON-SOAP-REST comparisons, and explain when is better to use SOAP:
Need for standards and stable contracts (see "PROS" section). PS: see a typical "B2B need for standards" described by @saille.
Need for tools (see "PROS" section). PS: standards, and the existence of formal verifications (see bellow), are important issues for the tools automation.
Parallel heavy processing (see "Context/Foundations" section below): with bigger and/or slower processes, no matter with a bit more complexity of SOAP, reliability and stability are the best investments.
Need more security: when more than HTTPS is required, and you really need additional features for protection, SOAP is a better choice (see @Bell, 32 votes). "Sending the message along a path more complicated than request/response or over a transport that does not involve HTTP", S. Seely. XML is a core issue, offering standards for XML Encryption, XML Signature, and XML Canonicalization, and, only with SOAP you can to embed these mechanisms into a message by a well-accepted standard as WS-Security.
Need more flexibility (less restrictions): SOAP not need exact correspondence with an URI; not nedd restrict to HTTP; not need to restrict to 4 verbs. As @TravisHeseman (9 votes) says, if you wanted something "flexible for an arbitrary number of client technologies and uses", use SOAP.
PS: remember that XML is more universal/expressive than JSON (et al).
Need for formal verifications: important to understand that W3C stack uses formal methods, and REST is more informal. Your WSDL (a formal language) service description is a formal specification of your web services interfaces, and SOAP is a robust protocol that accept all possible WSDL prescriptions.
CONTEXT
Historical
To assess trends is necessary historical perspective. For this subject, a 10 or 15 years perspective...
Before the W3C standardization, there are some anarchy. Was difficult to implement interoperable services with different frameworks, and more difficult, costly, and time consuming to implement something interoperable between companys. The W3C stack standards has been a light, a north for interoperation of sets of complex web services.
For day-by-day tasks, like to implement AJAX, SOAP is heavy... So, the need for simple approaches need to elect a new theory-framework... And big "Web software players", as Google, Amazon, Yahoo, et al, elected the best alternative, that is the REST approach. Was in this context that REST concept arrived as a "competing framework", and, today (2012's), this alternative is a de facto standard for programmers.
Foundations
In a context of Parallel Computing the web services provides parallel subtasks; and protocols, like SOAP, ensures good synchronization and communication. Not "any task": web services can be classified as
coarse-grained and embarrassing parallelism.
As the task gets bigger, it becomes less significant "complexity debate", and becomes more relevant the robustness of the communication and the solidity of the contracts.