Difference between geometry and $geometry QGIS

Right next to the list of functions your screenshot is of, should be the currently selected function's documentation. For me, with 2.18.12 those read:

$geometry Returns the geometry of the current feature. Can be used for processing with other functions.

geometry Returns a feature's geometry.

This is like the different of a method versus a function in object-oriented programming. $geometry is this/self's geometry. You can think of it more like an attribute or property maybe.

geometry on the other hand requires/allows you to pass a feature to it and it will return the corresponding geometry. As the documentation shows, you can use this to get the geometry from a attribute-based feature selection:

geometry(
  get_feature(layer, attributeField, value)
)

What else you can do with it is just limited by your creativity and the sources for features to pass to it. :)

$geometry returns the geometry of the current feature as in geom_to_wkt($geometry)

geometry returns the geometry of a specific feature as in geom_to_wkt(geometry(get_feature('my_layer', 'my_feature', feature_id)))

You would use the second case if you wanted, for example, to process the current feature against a specific feature:

intersects($geometry,geometry(get_feature(layer,attributeField,value)))