I'm just gonna start-off by saying that my post is referring specifically to competitive situations. Be it people comparing their progress in a non-PvP situation, or fragging each other directly, I feel that cheating, exploiting, and so forth only really matters when you're trying to compare your feats to that of someone else. If it's purely a single-player game and you aren't in any form of race (or whatever) with someone else, then do whatever you want. That said, my thoughts on bug exploiting are such:
It depends on the glitch and the situation. 'Nuff said for those who tend to go tldr on you, but I'll go into detail about what I mean for those who want a little more meat on their plate:
In some cases, exploiting a bug is considered 100% viable. Take for example Super Metroid. There's loads upon loads of bugs in that game which players can, and do, take advantage. However, it isn't really considered cheating when they do it because it's all in the name of speed-running and over-all just having fun with the game. Learning how to work the mechanics of the glitches is considered to be all a part of the challenge in doing a successful speed run. If you want to try and beat the current record for speed-running the game, you better be willing to practice some funky techniques. The bugs tend to be very specific in how they're performed (to the point where the average person has a one-in-a-million shot of accidentally tripping them), so it's actually harder to play the game using the bugs. Thus, it's considered a viable means of comparing your achievements to other players.
Then you have cases where a bug is both easily performed, AND makes the game easier in some way. When exploited in a competitive environment, I consider it to be whole-sale cheating. It can be a bigger deal in some games than others, but cheating is cheating. If I'm playing a pub game of Team Fortress 2 and someone exploits a glitch to kill me, then it's cheating. I'm not gonna throw a huge fit about it and it isn't a big deal because nothing's really at stake. At worst I'll just find a different server to play on (or stop playing the game until the bug is fixed if it's rampant). However, if I'm playing a tournament with my clan against another clan and they start exploiting a bug to win, then it's crossing a line. I still won't throw a huge fit about it (since my experience with trolls and exploiters is that they'll just do it more), but I'd certainly file a petition with the tournament holders to strike the match from the records due to cheating. If they don't, I'd find a new tournament venue to play from.
Then you have cases like the infamous banning of an infamous high-end raiding guild in World of Warcraft for exploiting the final encounter of the current expansion. I stand 100% behind Blizzard on their decision to do so. Not only is this a guild that's been known to exploit in the past and got away scot-free, but then they had the balls to exploit on the biggest and most anticipated boss in the game. On top of this, it's a well known rule in the ToU that you have to agree to in order to even play the game that knowingly exploiting a bug is ban-worthy. So the guild knew what they were doing. They knew how Blizzard could react. They did it anyway.
So yeah, for a tldr wrap-up: Some glitches take skill to trigger and can actually make the game harder to play than just doing it normally, where other glitches simply make the game easier. In single player I don't really care one way or the other. In a competitive environment, however, the former is generally acceptable because it's hard to complain about someone making the game harder on themselves and still beating you, where using the latter form of bugs to exploit the game is cheating no matter how you want to slice it. How big of a deal it is to use the cheats would be another debate entirely, but cheating is cheating.