The core argument used against Web Storage says because Web Storage doesn't support cookie-specific features like the Secure flag and the HttpOnly flag, it's easier for attackers to steal it. The path attribute is also cited. I'll take a look at each of these features and try to examine the history of why they were implemented, what purpose they serve and whether they really make cookies the best choice for session tokens.
The Secure FlagThe secure flag is quite important for cookies, and outright irrelevant for web storage. Web Storage adheres to the Same Origin Policy, which isolates data based on an origin consisting of a protocol and a domain name. Cookies need the secure flag because they don't properly adhere to the Same Origin Policy - cookies set on https://example.com will be transmitted to and accessible via http://example.com by default. Conversely, a value stored in localStorage on https://example.com will be completely inaccessible to http://example.com because the protocols are different.
In other words, cookies are insecure by default, and the secure flag is simply a bodge to make them as resilient to MITM attacks as Web Storage. Web Storage used over HTTPS effectively has the secure flag by default. Further information on related nuances in the Same Origin Policy can be found in The Tangled Web by Michal Zalewski.
The Path AttributeThe path attribute is widely known to be pretty much useless for security. It's another example of where cookies are out of sync with the Same Origin Policy - paths are not considered part of an origin so there's no security boundary between them. The only way to isolate two applications from each other at the application layer is to place them on separate origins.
The HttpOnly FlagThe HttpOnly flag is an almost useless XSS mitigation. It was invented back in 2002 to prevent XSS being used to steal session tokens. At the time, stealing cookies may have been the most popular attack vector - it was four years later that CSRF was described as the sleeping giant.
Today, I think any competent attacker will exploit XSS using a custom CSRF payload or drop a BeEF hook. Session token stealing attacks introduce a time-delay and environment shift that makes them impractical and error prone in comparison - see Why HttpOnly Won't Protect You for more background on why. This means that if an attacker is proficient, HttpOnly won't even slow them down. It's like a WAF that's so ineffective attackers don't even notice it exists.
The only instance I've seen where HttpOnly was a significant security boundary is on bugzilla.mozilla.org. Untrusted HTML attachments are served from a subdomain which has access to the parent domain's session cookies thanks to cookies' not-quite-same-origin-policy. Ultimately as with the secure flag, the HttpOnly flag is only really required to make cookies as secure as web storage.
Meanwhile, the unhealthy state of the cookie protocol leads to crazy situations where the cookie header can contain a blend of trusted and untrusted data. This is something the ill-conceived double-submit CSRF defence fell foul of. The solution for this is yet another cookie attribute: Origin.
Unlike cookies, web storage doesn't support automatic expiry. The security impact of this is minimal as expiry of session tokens should be done server-side, but it is something to watch out for. Another distinction is that sessionStorage will expire when you close the tab rather than when you close the browser, which may be awesome or inconvenient depending on your use case. Also, Safari disables Web Storage in private browsing mode, which isn't very helpful.
ConclusionAt first glance it looks like cookies have more security features, but they're ultimately patches over a poor core design. For an in depth assessment of cookies, check out HTTP cookies, or how not to design protocols. Web Storage offers an alternative that, if not secure by default, is less insecure by default.