Why are sabre bouts so fast and frantic while épée bouts can look slow and cautious?

Mostly because of the rules, not the weight of the weapons. Sabre rewards the fencer who attacks first and makes it very easy to land a hit, while épée gives the attacker no protection at all, so one weapon pushes you to go and the other punishes you for going without a good reason. An épée is heavier, under 770 grams against under 500 for foil and sabre, but a couple of hundred grams is not what slows the game down.

In sabre, right of way decides who scores when both fencers hit, and the one who starts a proper attack first has priority. You can score with the edge, the flat or the point, the target is everything above the waist including the arms and head, and light contact registers. The lockout, the window in which a second hit still turns on a light, is 170 milliseconds, raised from 120 in 2016, so an attacker who gets stop hit usually still has time to land and win. Because attacking beats defending, both fencers try to seize the attack the moment the referee says fence. Sabre also bans crossing your feet forward, so that attack comes as quick hops and a flying lunge.

Épée has no right of way. The whole body is target, the point has to land with real force (the tip must be pressed with more than 750 grams), and if both fencers hit within about 40 to 50 milliseconds, both score. That means a slightly early or sloppy attack is simply met by a stop hit to the wrist or arm, or by a double that only helps whoever is already ahead. So épée fencers spend time measuring distance, faking and testing reactions while waiting for a real opening. Plenty is happening, just in small steps and feints.

Foil sits between the two: it has right of way like sabre, but a smaller target (the torso), hits with the point only and a 300 millisecond lockout, so parries work better and bouts swing back and forth more. Speed in sabre and patience in épée are both learned skills. If you are choosing a weapon, try all three. Fencers who love explosive starts often gravitate to sabre, and those who enjoy a slow tactical battle often fall for épée.