Before watching the instructors video on the exercise, I was not aware of the option of unwrapping only specific faces (i.e. those in the current selection). I used only seams to support the unwrap operation and get a "good" result.
It strikes me that seam marking and unwrapping by specific face selection are somewhat interchangeable (at least for texturing). Is there any specific advantage or disadvantage to using one over the other (other than "working speed": I unwrapped the handle of the axe using the specific face selection to get the same result as seam marking in this video, and it just took way longer) ?
Good catch! They are basically interchangeable, except that marked seams stay with the mesh, while selection unwraps don't, so you'd need to select the same area every time which can be tedious. The one area where I like to use selection instead of (or in combination with) seams is when using the project from view style of unwrap. The advantage of that is it makes the UV's look exactly like what you see in the 3d view while a regular unwrap is sometimes warped, but the disadvantage is that there can be stretching if you're not careful.