What is the difference between a view mode and a style plugin?
I'll give this a shot.

A view mode is a concept that exists for entities. A node has for example the view modes Full content, Teaser and RSS.
And as you say different view modes can have different settings. An image might be just an URL in the RSS view mode and a large rendered image in the Full content view mode. It is also possible to have different order and visibility settings for each field in each view mode.

A Panel region or a Panel pane has something called a Style setting, this is also a Style plugin. There are styles such as No style, Rounded corners etc. Changing style will change the markup that surrounds your region or pane. In most cases the Style setting for the pane is something you do not change. 

But to answer your question, yes a region or a pane always has a set "Style" and that Style is by default set to use the style plugin "No style".

A style plugin is a concept that also exists in Views. Under the setting Format in Views you have the option to choose between Style plugins such as Grid, HTML list, 
Jump menu, Table and Unformatted list. When you choose a different style plugin the markup surrounding and in between each row changes.