Via using the
dynamic_sidebar() you can change/edit the widgets in the widgets area of the appearance settings. If you use
the_widget() you hard code it straight to template so you need to modify the code to change what is displayed.
Both have their uses. Too many sidebars tend to clutter the widgets area, but is very good for users that don't want to mess with the code (or if you swap around widgets/widget settings often).
The_widget isn't dependent on having a sidebar so it has it's uses too if you have a lot of custom widgets that you use around your templates, like in the footer for example.
But I don't think there are straight up benefits of using one vs the other, other than usability wise.
EDIT: altough I do agree with Pieter, I think there are at least fringe cases where you can get use out of
the_widget(). For instance a project I'm working on currently I have made a custom RSS widget for displaying icons according to network wide article types. (We have multiple sites which are all linked). We also have these RSS feeds on the bottom of the page that show content from our other sites and I see no use in coding a custom function with the same functionality when I already have a working widget.
And yeah, I could use a sidebar to show these widgets but as we have multiple people updating the sites and not all are so tech savvy, I figured it was best the sidebars area had basically just the containers that they need to edit.
So I used
the_widget to avoid duplicate code and to keep interface lean to other admins. But as well said by Pieter, you are probably better off always using wigdets with sidebars as they are intended to.