There's a couple of differences to me. One is that Heenan wasn't half so aggressive about it as is Cole. Heenan would deride faces for their heroic actions or whatever it was the crowd was cheering, but he rarely completely dressed them down at every level. Cole, meanwhile, has done that to guys like Daniel Bryan, for example. When Cole got on Bryan's case back in the day (he's toned it down somewhat) he was vehement about destroying any shred of credibility that Daniel Bryan had. But this wouldn't be so bad if it weren't for the other factor - there's no one mitigating Cole's influence. Jerry Lawler's a pathetic excuse for a color commentator who can do no more but half heartedly mumble "Oh, come on Cole" and make fat jokes. Josh Matthews is spineless and if he's not doing straight play-by-play he has almost no idea what he's doing in the booth. Booker T is the best thing Cole has to a foil, but even he rarely counters Cole well. Michael Cole, if nothing else, is an excellent verbal debater and he completely dresses down his competition. Heenan was the same, but the difference was that he had Monsoon in the booth to take him to ask for it. Another factor is that Cole is the play by play man - he is, quite literally, the voice of the WWE. Heenan was just a heel who sat in the booth. It was Monsoon's job to call the action, to sell the product, to talk about advertisements and upcoming events and so on. Monsoon was the voice, not Heenan. What we have today is the voice of the WWE, the most dominant figure in commentary, shitting all over guys that are trying to get over. This is fundamentally different from what Heenan's role was, which was only to provide color commentary. The final thing is that Cole was never a wrestling character who had already garnered heat and the audience decidedly hated. Cole made a slow, subtle turn to the heel side. This is subtly different from Heenan. Heenan was a character that everyone hated. Cole isn't so much of a character, and because of his penetration into reality, his words have a different effect.
What it all boils down to is how heavily tipped the scales have become to Michael Cole's heel commentary. He dominates every aspect of the booth. He's more vehement than Heenan was, he gets more time to speak, he dominates his competition, he has the force of the play by play role, and he has more reality penetration. The net effect is that when Cole decides to tear something apart, he does so with remarkable success. And because no one ever takes real retribution on him (unlike Heenan, who was frequently taken to task in the ring) he can severely destroy someone's credibility with his actions. He definitely impeded the success of Daniel Bryan more than he helped him, and has done the same to others.