Also explained in this video.

TimThumb

• Images are resized on-the-fly
• Can resize images not uploaded with WordPress media uploader
• Does not create multiple images on your server (storage)
• Uses cached image(s)- some users prefer this rather than having to load same image in multiple sizes on a page
• Easier to change width & height of an image (works out-of-the-box in most themes without any setup required)
• Does require ability to write cache images on server (folder permissions)

WP Post Thumbnails

• Image sizes have to be pre-defined in theme
• A new image file is created and stored on upload to your server (if you have server storage limits, this is not ideal). 1 image can result in many images on server.
• If you ever want to change an image size, you’ll have to regenerate thumbnails for all previous posts.
• If you have existing posts, they will need to have thumbnails generated otherwise site will not have images in required sizes
• If you switch themes, it’s likely that the images will not be sized perfectly for the next theme