I notice that some of the pics of the refueling boom have a basket, some have a probe, and some have a probe with wings...how many different fittings do you have?
The KC-10 has a boom,
centerline hose,
and some have wing pods.
The 135 has a boom, a strap-on boom drogue adaptor (the hose looking thing strapped onto the boom), and some have wing pods. And I'm too lazy to look up their crap
The boom can pump gas more quickly (~ 9,000 PPM for the boom vs. ~ 2,000 PPM with the hose), and is safer, since you don't have receivers always losing their probe in it, taking the basket with them (sorry 107...), or taking 5 minutes to get into contact because of a little light chop. But it needs a much bigger plane, and a separate operator instead of just flipping a switch and letting the other plane do all the work.
Both have their pluses and minuses, but it really comes down to which one is better for each service