The difficulty comes for me when I see an understudy or a replacement too early: before I've reached that magical point when the show stops "transporting" me. I dealt with this when I went to see the replacement Tribe in HAIR, for example. Even though I saw the original cast five times, I still fell head-over-heels into hippie-land every time I entered the Al Hirschfeld theatre. I became a draft-dodging, hash-smoking, inspired flower child -- not a critic. That meant that when I went back to see the new "Starshine" Tribe, I was caught off guard. No matter how great the new cast was, I reacted emotionally as if they were impostors. This Claude looked vaguely like "my" Claude... but it just wasn't him! This Berger had similar hair and energy to "my" Berger... but something wasn't quite the same! Interestingly, Diana DeGarmo as Sheila was the most successful (by which I mean least distracting) replacement lead in that show, and I think it has to do with her physical dissimilarity to Caissie Levy, who originated the role. Because they look substantially different, my emotional brain didn't suffer the same confusion, and I enjoyed her performance the most out of all three leads in the new HAIR Tribe.
Then I consider a show like Wicked, which I have seen three times, with three different sets of leads (Megan Hilty/ Shoshana Bean; Kate Reinders/ Ana Gasteyer; Annaleigh Ashford/ Stephanie J. Block). In this case -- perhaps because the performances fell over such a vast time period (four years) -- I had no problem accepting each actress in her role. Had I seen original leads Kristin Chenoweth and Idina Menzel multiple times over their reign as the Witches of Broadway, I might have had a harder time emotionally accepting their replacements.
I guess my appreciation of replacement casts on Broadway comes down to the way I relate to the characters, and, in fact, each specific show itself. When I am watching for the clues to the craft of theatre, and can pay attention to the slight variations that new actors brings to their roles, I am totally thrilled, and I thoroughly enjoy discussing the minutiae with my other theatre-nerd friends after the performance ends. When I am emotionally involved with the characters or storyline to the point that I "forget" I'm in a theatre, I can find the replacements distracting. (This is why I'm glad I never saw an understudy go on in Passing Strange or [title of show], even though it would have been interesting from the point of view of craft.)
I must point out that an actor's status as a "replacement" only really matters if you've have seen a different cast before. For fans who never saw the original cast of HAIR, Starshine Tribe stars Kyle, Ace and Diana WERE the originals, the ones to whom all future comparisons would be made. "My" cast of RENT, for example, was the First National Tour cast; it was a huge thrill for me when, 10 years after my first experience with RENT, I saw "my" Angel (Shaun Earl) play the role again on Broadway.
I regularly get emails and tweets from out-of-towners who ask, "Is it worth going to see Next to Normal now that Aaron Tveit is gone?," or "I can't believe I missed Angela Lansbury in A Little Night Music -- should I still see the show?". My answer is always YES, go see it!! Stunt casting aside, actors don't make it onto Broadway without being amazing. Never skip a show you want to see just because the original cast is no longer there.
If you have the chance to see a show multiple times, I recommend checking out different leads. The intricacies will teach you a lot about the living nature of theatre (How does one actor react to a loud sound cue differently from another? How does an on-stage couple's chemistry differ when an understudy is on in one of the roles?). If you have a strong connection to one or more of the actors you've seen the first time around, be prepared for a bit of emotional confusion at first; but try to put it aside and appreciate what the new actors bring. It's a different way to watch a play, but one that will allow you a new way to celebrate what makes live theatre amazing.
Now tell me: Who are your favorite Broadway replacements? Any you thought were better than the originals? Any you'd give anything to forget?