IMO the debate is less Weber vs Hedman, more Weber vs Laperriere.
Norris finishes
Laperriere: 1, 2, 4, 4, 5, 5, 8, 9
Weber: 2, 2, 3, 4, 4, 6, 8, 10
That's really, really close. Add up the numbers and you have 8 seasons each totaling 38 and 39 (this isn't a meaningful metric, just shows how close they are).
Peak/Prime
If there's a difference, IMO it isn't appreciable. Weber's peak may have lasted a bit longer, as Laperriere's prime was cut short by a broken leg in the semifinals.
Longevity
Weber has a clear longevity advantage, playing from age 20-35+ compared to Laperriere's 21-32. Again, Laperriere's was cut short by a major knee injury which forced his early retirement.
Playoffs
I have a feeling that playoffs will ultimately be the arbiter between them, and that should be an advantage to Laperriere. His legacy gets dinged for the fact that he missed a couple of good playoff runs with injuries, but bear in mind he did participate in some very successful runs of his own.
1967 - Full disclosure, this was a poor showing. Laperriere hadn't played playoff hockey since his injury in the '65 semis, was coming off probably the worst season of his career, and struggled with penalties and gaffes.
1968 - During Montreal's first-round sweep of Boston, two different papers (the Montreal Gazette and Ottawa Journal) said Laperriere had played the best hockey of his career.
The following series was marked by a scoring explosion from JC Tremblay (6 points in 5 games) but Laperriere also had a very good series, only on the ice for 3 of Chicago's 10 goals despite playing heavy minutes.
Not to read too much into this, but when the Gazette's Pat Curran rolled out the annual Conn Smythe debate (an article where he actually boosted for Dickie Moore to be considered), he mentioned Laperriere second after Gump Worsley front-runners. In the end, Glen Hall won the award as the rare losing-team selection, but I have always seen Worsley mentioned as the "snub". If Curran meant his list order to be taken literally, one could presume that Laperriere was the top Smythe candidate among skaters.
Noteworthy: After the Cup was won and award results released, Curran commented: "Those who have followed the Canadiens closely should also be surprised at the fact that... J.C. Tremblay had 26 points to only 12 for Jacques Laperriere. The latter was by far the best Montreal rearguard all season."
1969 - This is widely remembered for Serge Savard's big run that ended in a Smythe. But Laperriere was also very good during that run, and arguably had more of a hand in Savard's success than we give him credit for. He was especially good during the Habs' semifinal series against Boston (the Final was a cakewalk over St. Louis, so the semifinal was effectively for the Cup). He scored a key goal in the Habs' Game 5 win to take the series lead, and was arguably even better in their series-clinching OT win in Game 6.
Habs coach Claude Ruel on Laperriere's Game 6 performance, which effectively won the Cup for Montreal: "Listen, Jacques Laperriere played the best game of his life out there tonight and no one ever hears about him or Harris. I don't hear anybody going around shouting to make them all-stars. And I don't know what they have to do to be all-stars. Laperriere did everything tonight. He blocked shots, he skated, he killed penalties, he was on the power play."
Serge Savard describing one of the key goals that earned him the Smythe: "On the goal I scored I told Jacques (Laperriere) to stay back because I was going in front of the net. I went in close and when Yvan Cournoyer had a chance, the puck came to me and I shot it from close range." (noteworthy: in this same game, Boston scored the tying PP goal when Savard accidentally cleared the puck directly off Laperriere's butt, pinballing it back into scoring position. Savard actually played forward on that Habs PK unit, with Laperriere playing back)
Subjectively, Laperriere's name tended to be mentioned before Savard's when the writer wanted to emphasize the Habs' stingy defense, and vice versa when emphasizing their scoring power.
1971 - Bear in mind, this was the season that Serge Savard was lost in the regular season with a broken leg. As luck would have it, a young Guy Lapointe stepped in and replaced him almost seamlessly -- until the playoffs, when Laperriere stepped up to showcase as the Habs' best two-way force.
Laperriere had 4-9-13 in 20 games with a +11. Compare to Tremblay's 3-14-17 with a +5, and Lapointe's 4-5-9 with a +7, all three with the same GP.
He again was a pivotal figure in the Habs' 7-game series against Boston, a monumental upset that prevented a Bruins dynasty.
This was after he missed nearly half the season with a pinched nerve in his neck that, at one point, paralyzed his left arm. He returned in time for the playoffs, then broke his wrist in the first game of the Final, and played through it.
In the Montreal Gazette's spread on the Cup-win, it applied the word "hero" to three players: Henri Richard, Ken Dryden, and Jacques Laperriere.