There is no "one way is better than the other"; it really depends on your current needs. You need to be sure that you are using the tool and method that does the most work for you to get the effect you want with the least amount of effort. I suggest you do some more reading and practice with the Pen tool before settling on a methodology. The Pen tool is the core tool of Illustrator and really makes Illustrator what it is.
The first example offers complete control over the direction of the line while maintaining consistency in width, and that includes bezels in the corners. Consistency in lines is really important in graphic design. Even though style is an "all or nothing" proposition with an open path, that is really all you need (or want) more often than not.
But the open path is especially useful if any one of your path components all of a sudden needs to be curved, because that's where the real power of bezier paths come into play. A closed path is almost impossible to keep consistent in that scenario. If you create path and then play with the Anchor tool, you'll quickly see what I mean.
In your second example, it is harder to maintain path width consistency in a straight or curved line. You get a lot more control all of the individual components of the line, but it can also mean a lot more work when an open path would have been sufficient. An easy way to get from the first to the second example, however, would be to create a path that is generally what you want it to be, then use the Outline Path tool to give you something that has more control, but there is no going back once you've done that.
In my work, I would really only go with the closed path for a line if more of a casual, almost hand-drawn look is required for a project. Otherwise, I just stick with the open path for just about everything.