Given that they share a common ancestor, the general contours of rugby and American football are quite similar. The goal of both sports is to get the (oval) ball into the opponent's end zone (try or touchdown – most valuable score). In both sports it is also possible to kick goals. In the specifics of the two sports however there is much significant variance, e.g. in rugby it is always illegal to throw the ball forward whereas in American football one forward pass per play is allowed.
It is not my intention to compare and contrast the (relative) merits, or otherwise, of rugby and American football generally, but rather to investigate the 'efficiencies' realised by the approaches to specialisation current in the respective codes. Aptly enough, given that Scrum got its name from rugby, we can view rugby teams as analogous to feature teams and American football teams as analogous to a collection of component teams:
In (professional) American football it is common to have an offensive team, a defensive team and a specialist team (e.g. for kicking plays). There is very little overlap in the personnel of these teams, a quarter back never lines out in defence. As a consequence, no one team can effect the complete set of possible plays. As such, when the ball changes hands in American football, play usually stops to allow both sets of players to be replaced.
An obvious advantages in this is that, for example, a defensive tackle can concentrate on breaking the offensive line, and achieve dizzying excellence in that one aspect of play. An offensive tackle will specialise in blocking players attempting to get at his quarter back, expending very little effort on learning how to break lines...
There are of course specialist roles in rugby too, roughly speaking a rugby team is split into backs and forwards, whereby the forwards contest and control the ball and the backs move the ball. Two special positions are further instrumental in directing play and providing a link between the forwards and the backs; the scrum half and the fly half. The fly half also usually takes care of all of the place kicking.
Rugby players therefore have very specific jobs to do but are regularly called upon to help the team effort in ways not defined by their role (generalising specialists). One of the advantages to this approach is that players can improvise; should the fly half receive the ball during a play designed to provide a goal-kicking opportunity, wherein he sees the envisaged opportunity blocked, he can still choose from a range of possibilities, including options beyond his usual remit, to further the play (sometimes to even greater advantage than originally planned).
Assuming that quality at the highest levels in both sports is comparable, and that points scored are a reasonable measure of 'value' delivered (to all parties; participants, fans, investors, etc.), we could define efficiency as follows: points scored/(play time * squad size)
The average number of points scored in NFL games through the 2008 season was 44 (source) A similar figure for rugby games is 38 (source). In a single game of American football (60 minutes play time) up to 90 players may be actively involved, although there are only ever 11 players per team allowed on the pitch at any one time. This yields a rough scoring efficiency of 0,008. In a rugby game there are a maximum of 44 players involved (both teams start with 15 players and can make up to 7 substitutions), and play time is 80 minutes, which results in a scoring efficiency of 0,01 – 25% greater than the throughput for American football.
If we consider that an American football game usually takes about three hours to complete (due in part to the regular substitution of entire teams), compared to a rugby match which normally completes in well under 100 minutes, the efficiency gap yawns even wider... In that case the feature teams (rugby) could be delivering value at up to 4 times the rate of the collection of component teams (American football).