While the conventional wisdom before avios was as stated above, post-avios now requires segment pricing and eliminates the multiple stopover greatness of BA miles. Additionally, RT from NYC to LIM is not 40k for J as you stated, but now 80k for J ... and with no stopovers.

For me, by far the best use are short routes with high prices, such as Lima>Cusco>Lima. RT tickets can be $1000 for coach, but only 9000 miles.

While I understand there may be few best use scenarios that are all-encompassing, most of the best uses vary between individuals as stated by previous posters.

For me ...

BA is used for short hops or positioning with other longer itineraries.
WN is used for domestic trips in conjuction with a companion pass.
Saving Delta for an Africa trip
AA for a RTW trip
CO for mini-RTWs
etc ...

Maybe we could start a thread/sticky where FTers highlight their individual best uses of different programs as a means of new FTers learning what is and isn't important to them?