My biggest one was the disappearing player. I recruited some players from a forums I used to hang out at, and in the second or third session, one player completely disappeared. He still posted on the forums.
In a later thread discussing pet peeves on those forums, several people mentioned that most online games crash and burn. They said many players simply quit without informing anyone, some much so that it was acceptable behavior.
That sort of killed my desire to run more online games.
The Laundry machine noise is one annoying problem (A player is doing his laundry during the game and the noise is louder than anything else)
The mysterious connection problem that appears 5 minutes before game start.
@Tormsskull
This is actually how I got started as a GM, the guy who recruited the group never showed for the first session and disapeared completely so I told the group : "Someone else should be the GM"
There are a number of specific issues with online gaming, as well as a number of strengths. Since you asked about the weaknesses and issues, I'll focus on those.
The type of issue depends on the online medium, and whether it involves a specific group of friends or a broader audience. Is it a persistent world, or by-campaign? I'll cover 10 general issues in this post. If anyone is interested in something more specific, please feel free to ping me.
1. Timezones are a big one...both in understanding them, and because you're also dealing with the separate scheduling of people across the globe. There are two "solutions" to this.
The first is to be somewhat flexible on your starting time...say give it, unofficially, a half hour. Be forgiving if someone shows up a little later than that. Just work them in as best you can. Importantly, remind your players to keep you updated...give them a way to contact or message you if they'll be late. If someone is consistently an hour late, speak with them about it, but if it's an occasional happenstance, just let it go. This is especially true with adult players; it isn't like when we were kids, or in college, anymore.
The second is that for one-off scenes, keep a waitlist. If someone is a half hour late, then they lose their spot. This lets you keep the scene full and keep moving, while being fair to everyone involved.
The third is accepting that scheduling conflicts, especially online, happen. Work with it as you can. Be upset at the constantly-late player if you need to, but by using the tactics above then chances are you're more able to work around it.
2. A second perhaps less obvious issue with online gaming is writing style. Players and DMs are so particular about style in TT and this does not go away when you are online. For example, sometimes players or DMs might insist that character responses are typed with a specific form of indentation. Other players may write short responses or none at all.
What can work here is either being accepting of a variety of styles (you may need to remind your playing group(s) of this from time to time. Another is to set an example of a preferred format and to work from there.
This is of course moot if you're playing over mic.
3. Mixed play styles and managing arguments. Somewhat different than #2, this refers more to play style...the old rollplayer versus roleplayer debate. Chaotic versus law. Some of these debates can become more intense online, where the screen hides us, or hides the very real impact of the other people at the table around us. Don't be afraid to enforce some time away from the table, or ask for a topic change. Don't be afraid either, to speak with players if you need to about the type of environment that's appropriate for the game. Sometimes, we all need a reminder to step back somewhat.
Managing online arguments could be its own point...and will be in just a moment.
4. Managing online arguments, the expectation of privacy, free speech, etc. Sometimes arguments become worse online. Also there are areas that people will explore online that they may not in life. For example, someone may come into your forums and begin arguing about gun control. They might, for example, joke about assaulting a public figure. Aside from being potentially jarring to the game environment, some of these statements can place you in an odd legal position (especially as online tends to = easily logged).
In the end, and this is important...think of the online space where you run your games as a home. You have the authority to set rules for it. This is not any different than Paizo having rules for its forums. If you run an online game, your policy for handling these behavorial situations is going to be important.
Also important: the policy does not need to be long. Some of the most successful games I have seen possessed very simple policies that amounted to: don't be a jerk.
5. Easy start/well-organized documentation becomes important. This is more true with a persistent online world, but also with a smaller one. Make the documentation easy to understand. Provide a "jumping in" document to help your players get on their feet and playing. That's what they're there for, after all.
6. The underbelly. Gaming can be an empowerment fantasy. Online gaming can also be a means to explore concepts and ideas you may be uncomfortable doing so in life. However, there are some fantasies that may not fit your style of game you're hosting, running, playing in and so on. Be aware of this and be upfront about it. If you've never heard or had someone ask to play a half drow hybrid catperson dominatrix elf half vamypre half sorceress...bless you.
7. Burst abilities and powers can become all-day powers. At tabletop, we become used to being able to do so many encounters in a session. Online, encounters simply take longer. This causes a few issues. The first is that it can make the adventuring day shorter. This greatly enhances the value of burst abilities, limited abilities, and so forth. The second is that it will or can skew balance. Classes that were balanced around a limited use of their abilities suddenly aren't.
8. Combat takes longer. This is important enough that it's worth repeating a second time.
9. Communication is the source of most conflicts; online, this can be magnified (unintentionally). For this reason, never be afraid to ask for clarification. If your players are new to online play, or even if they aren't, encourage this practice. It can save you some headache in the long run and encouraging a better gaming experience in the longer term.
10. Remember you have DM fiat (except in the case of PFS). There are many differences in online and offline gaming, and not all of them are evident at first glance. Some are more subtle than others. Don't be afraid to make some adjustments to enhance the experience if something doesn't quite work as well...or just needs a minor tweak.
Relevant Experience: About a decade's worth of assisting in running online d20 massive worlds/games, with as many as 4 scenes/day. You are all good people. :)
I fibbed. There needs to be 11:
11. Different comfort levels, and levels of privacy. Privacy is more of a thing to some players than others. This is less so in the d20 medium, but something to be aware of regardless. If you can, provide players options for how much they disclose, and when.
My #1 problem is beginning a monologue or describing some important scene (using google+) and then losing my connection partway through without realizing it.