Flash vs AJAX
The main commonality between Adobe Flash and AJAX (Asynchronous Javascript and XML) is their ability to add interactivity to web pages, thereby improving the overall web experience of the visitor to the site. The main difference between the two is in how they achieve their goal. Flash is a complete platform that provides all you would need in a neat package. On the other hand, AJAX is just a collection of preexistent web technologies that include XML, HTML, DOM, CSS, and Javascript. With Javascript being the glue that holds them all together. In order to make use of AJAX, you need to know XML and Javascript at the very least.
Comparing the two, there is wide berth when it comes to their capabilities. Flash is able to draw on screen and create complex graphics, something that is beyond the capabilities of AJAX. Because of this, Flash is the platform of choice when creating games, or other web programs that are very complex. Video is also another forte of Flash with YouTube being the biggest site that uses Flash to deliver videos to end users. AJAX is not able to do all this as its capabilities can only extend to the limits of the individual components. Basically, in terms of on-screen rendering, AJAX is limited to what XML can do.
When using Flash, you can always tell because it would occupy a certain space on the screen that no other element on the page can take. AJAX basically works on the background and is rather invisible. The only clue that AJAX is being used is when parts of the page change without reloading the entire page. That is basically the main goal and purpose of AJAX.
An example of a site that uses both Flash and AJAX is the very popular site Facebook. It uses AJAX in order to change the contents of the page; like when you go switch between home and your profile page. AJAX is also used in the notifications where you can get your messages or accept requests without having to leave the page you are on. Apps, especially the games, are a bit more complex, so Flash is used for them.
Summary:
Flash is a complete interactive platform while AJAX is a collection of web development technologies to add interactivity
Flash is able to play videos and draw graphics while AJAX is not
Flash does not integrate as easily to webpages as AJAX can