I'm developing a Chrome extension and I need to store some data and then get it in some point. I did investigation on available
It depends entirely on what your Chrome Extension will be doing. window.localStorage is HTML5 storage. Unless you're running it in the background page, it can only allow you to get and set data into storage for a specific domain. This is also true for code injected into the DOM, since it would use the localStorage on the web page.
In other words, you won't be able to share data across different web pages unless you use localStorage in the background page, which operates independently of web pages, since it has a chrome:// URI as its domain.
chrome.storage.local, on the other hand, is designed for Chrome Extensions and Chrome Apps to store data in a more central location. Since this isn't accessible to normal web pages, each Extension gets its own storage. One possibility is for your background page to handle dealing with the setting and getting of the data, while your content scripts deal with modifying and interacting with the web page.
However, these API's work in content scripts as well, and both of the extensions I've written use chrome.storage.local called from the content scripts.
As an example, I built a Stack App that preserves inbox items in Stack Exchange until you've actually read them, called StackInbox. Since Stack Exchange sites span across hundreds of domains, I chose chrome.storage.local because I could save the user's accountId and reuse it across all the sites, ensuring that the inbox data is synchronized, while also using this directly in the content script.
As a simple test, put some data in localStorage on one domain, in a content script, and try to pull it from another, and you'll see that the data won't be there. With chrome.storage.local, this isn't a problem.
Lastly, Chrome Extensions and Chrome Apps are whitelisted, since the user chose to install it, so they typically can do more things than a normal website. For instance, by specifying the "unlimitedStorage" permission in your manifest, you can store data well beyond the 5MB limit placed upon HTML5 localStorage.
For more information, see Google's documentation on Chrome Storage.