After the USAF flew Navy aircaft for several years (The F-4 and A-7), they learned to really like having an arrestor hook.  The F-15 and F-16, although totally non carrier capable, both have an arrestor hook, as does the F-117 and F-22.  Why?  Almost all military bases worlwide have "field arresting gear".  Theses system cannot stop an aircraft in the few feet at full touchdown speed the way carrier arresting gear do, but they can stop an aircarft that has landed and has already slowed down to keep it from running off the end of the runway, and that's how USAF uses it, to prevent their aircraft from running of the end of the runway in the event of some problem.  Maintaining the arresting gear is way way cheaper than fixing an aircraft that has overrun the runway.

And you're right, it does no use a rope, but a steel cable.  A rope would not begin to work for this kind of application.

Hope this answered your question.