The Quotations module is used to quickly and easily add styling to quotations that you want to appear on your site. It includes default sets of images (5 different sets of semi-transparent PNG quotation marks, both light and dark versions) and CSS that you can apply to specific CSS classes that appear on your site. Quotations are most often built in Drupal blocks.
Two common uses of the module are to 1) manually create a custom block that contains a quotation on the blocks page and add some CSS wrapper divs to affect the styling of the block or to 2) create a block using the Views module and then use the Quotations configuration page to adjust the default display of the Views block.
Differences Between Quotations and Other Modules
The Quote module depends on the core Comment module and provides a Drupal filter that allows a special [quote] tag to be used within comments on the site. It applies some very basic CSS styling to the quoted material. The assumption of the Quote module seems to be that the user wishes to comment on something someone else has previously written on the site either within the body of a node, or within a thread of comments.
The Quotes module allows users on a community-driven site to maintain their own lists of quotations and can be used to display blocks of quotations throughout the site.
The Quotations module doesn't have any dependencies and doesn't include any input filters. It assumes that the administrators have some existing quotations that they'd like to appear on their site as an extension of the content (for marketing purposes, sharing quotations of the day, etc.). It also provides a means for them to style their quotations using some built-in image sets and CSS files. It actually builds a custom CSS file for the admin within his/her "files" directory, based on how he/she configures the module.