So after hearing the cover only once as basis of your informed opinion, was there a susbtantial difference between James and Geddy's singing of "Paradise" and "Escape these caves of ice"? I am focusing on these two because these are the subjects of most criticisms.
I've now listened to it several times.
The word "paradise" appears in the song three times (or five times, since the middle time is "paradise, oh paradise" and the third time is "paradise, more than paradise" (at least that's what it sounds like to me)). The first time is around 5:13 of both the original and the cover. Is there a "substantial difference"? I would say so. Geddy and James don't sound much alike.
The second instance of "paradise" happens around 6:44 of the original and slightly thereafter on the cover. The phrasing of "paradise, oh paradise" is similar, but there appears to be a badly done overdub in the cover. When Geddy sings this line, it sounds effortless even though it's a very high vocal. James, on the other hand, is clearly straining to hit the notes, so there most certainly is a "substantial difference" between the two (as well as the lines leading up to it). This is not a criticism. As more than one person previously noted, Geddy was 23 years old when he recorded the original, whereas James was 30 years older. Maybe James could have hit the notes with the same effortlessness as Geddy when he was 23, but no one expects that from him now.
"Escape these caves of ice" occurs at 8:50 of the original and is followed shortly thereafter by the third instance of "paradise." As with the second instance, Geddy sounds effortless, while James sounds strained. James also appears to put more "oomph" into this one, which I assume was a stylistic choice. Again, there is a "substantial difference" between Geddy's and James's versions.
Would it be better if James had done it a little closer to the original or completely re-done the melody line to make it easier on himself? Maybe. Could he have sounded smoother? Under the circumstances, I think probably not. Do I like it as much as the original? No, but it's a really good cover, and I like hearing bands I like cover songs I like, so good job.