Aerobic vs Anaerobic
Climbing power is a combination of Anaerobic Work Capacity (AEC) and the upper limits of sustained aerobic power such as Onset of Blood Lactate Accumulation (OBLA), Lactate Threshold (LT) or Functional Threshold Power (FTP). Basically being able to ride at the highest aerobic level possible (L4 in the Fitlab zones) combined with the amount of work that can be done above that level (or shorter burst of high power output L5-L7 in the Fitlab zone). Since we are talking about climbing, it is relative power (watts per kg) that is more important than absolute power (watts).
Pro Comparisons
Table from: Critical Power and Anaerobic Capacity of Grand Cycling Tour Winners by C Dauwe
How It Pans Out On The Road
Every rider hass a different mix of these two skills and therefore responds to races and training differently. Some riders have high OBLA/FTP power to weight and therefore prefer to ride at a high aerobic pace with little or no bursts into the anaerobic range. While others may have less OBLA/FTP power to weight, but high AEC (volume of work they can do anaerobically). These riders will try to break up the rhythm of the other riders by doing high repeated anaerobic bursts (0-3 minutes), and settle back to their aerobic level. Their average power up the climb may be similar, but they go about it totally differently and require different amounts of recovery (the more anaerobic riders require more recovery and each effort is like burning a match, from a limited supply). One climbs steadily while the other breaks up the tempo with bursts of speed.
Wiggins/Indurain/Demoulin would be examples of the steady riders while Rodriguez/Pantani/Valverde use their AEC to break up the steady pace. The shorter climbs favour the riders with effective AEC, while the longer climbs suit the riders who prefer the steady, aerobic tempo. At the extreme end of this we can see the short 1-2 minute climbs in the Tour of Flanders suiting riders with a great deal of anaerobic power (Gilbert/Kwiatkowsk/Sagan). While at the other end we see Team Sky setting a high steady pace on long climbs to aid their riders with big aerobic engines. This fast, even pace also forces the more jumpy climbers to be put on their aerobic limit and reduce their ability to repeatedly attack.
Conclusion
What we learn from this is we need specific training depending on the profile of the event we want to do well in. We also need to look at selecting races suited to our unique abilities.
