The introduction of WAI-ARIA has added new ways for website owners to make sure their websites live up to the success criteria in WCAG 2.0. It is one of many tools that can help you create and maintain accessible web content.
In many cases normal HTML can be fully accessible without use of WAI-ARIA.
An example is the WAI-ARIA landmark roles (e.g. role="navigation"). These can be applied to sections of a page to indicate the semantic role of each section of the page. The same information can be provided with HTML5 sections (e.g. <nav>) for navigation.
The target group for WAI-ARIA is quite narrow. Blind people who use screen readers to access web content, and people using speech instructions to control their computers will benefit from WAI-ARIA. Most other users will not benefit from WAI-ARIA, which means that it shouldn't be your only means of creating an accessible website.
WAI-ARIA really shines if you have some non-standard functionality that is dynamic or interactive. In these cases, WAI-ARIA provides tools for making your non-standard functionality accessible for users of speech-based interfaces.
If you are redesigning your website from scratch, accessibility through native HTML should be preferred where possible. However, if you are retrofitting your site to improve accessibility, using WAI-ARIA might be the best way to get the job done in many cases.