What are the implications of different button sizes in terms of what the designer is trying to convey to the user and what the user is implying as to the function of the button?
For example, does a huge "Download" button convey anything different to the user versus a small button or is the button simply bigger to make it more prominent and get more clicks?
I suspect button size may sometimes correlate to the complexity of the corresponding action but I don't have any evidence of this other than my own observations. It does seem that large "Download" buttons tend to be direct download links as opposed to a link to the download page.
This will depend largely on the context where the button is applied. You'll actually find that in download websites the larger and brighter download button isn't always a direct link but may be advertising or affiliate links because it is usually how they generate revenue. On standard websites or desktop applications, the button size is generally related to the importance or frequency of the function or task to be performed. Users don't like having to go look for the important or frequently used features. Larger buttons also allows the incorporation of images with more detail, or two differentiate between similar images.
The use of ribbon layout design in MS Office 2013 applications is an interesting study in how the size, shape and layout configurations can lead to a different user experience to the traditional file menu and right click workflows. On web applications where designers also have to think more about mobile and smart device usage that incorporate touch and gesture input, the size of buttons then becomes a very practical issue because even though a larger button is easier to press (correctly) it also takes up valuable screen space.
I would treat it in a similar regard to typography. As in, there should be a natural hierarchy based on the UI and the relative content. Buttons should be a sufficient size to be seen and used, especially if they are integral to functional flow. CTA buttons will benefit from a bigger size though, depending on the overall design, colour selection may have a similar affect.