On my previous websites, I used to use a cookie to display a pre-home page only on the first visit. That worked great (see for example here), but using cookies is not so tre
The user will not know if you are using localStorage or a cookie. If a user disable cookies, localStorage will not work either.
There is no noticeable speed difference between the two methods.
sessionStorage is only for that browser tab's session. If you close the tab, the session will be lost and the data will be lost too, it's similar to a session variable on any backend language.
localStorage will be available for any tab or window in the browser, and will exist until it is deleted by the user or the program. Unlike a cookie, you cannot setup expiration. localStorage has a much larger storage limit as well.
You can use a Modernizr to verify if localStorage is available and if not, use store a cookie instead.
if (Modernizr.localstorage) {
// supports HTML5 Storage :D
} else {
// does not support HTML5 Storage :(
}
You can also forego Modernizr and use the check typeof Storage !== 'undefined'.