What are the pros and cons of internal gears?

what are the trade-offs between the internal gear hub and an external gear hub and derailleur?

The trade-offs are many and clearly in favor of derailleur gearing.

For example:

• Some internally geared hubs have "freewheeling" positions between gears so if the cable is not adjusted, you can end up with no gear, causing you to fall if trying to pedal hard while expecting to be in gear
• Repairing a flat is difficult. You need to remove the reaction arm and cable, and there is no mechanism to keep the chain in the correct position and tension it automatically. You also need to adjust the chain tension manually. In contrast, with derailleur gearing you just drop the wheel and put it back in after repair. No chain lube stained hands with derailleur gears. Typically there is no quick release either.
• Like "modern" STI shifters, an internally geared hub is in general not worth it to repair. Changing a hub is difficult and requires wheelbuilding skills, unlike changing an STI shifter which is simpler.
• Efficiency is lower than for derailleur gearing. For example, for 40000 km distance at 20 km/h (2000 hours), 2% lost power loses 0.667% speed (due to third-power-law effect of air resistance) and thus 13.3 hours.
• They may not withstand as much torque as derailleur gears. For example, if I stand on the front pedal with my whole weight (114 kg), pull up from the rear pedal with 25kg (25 + 25 kg) and pull up from the handlebars with 30kg, I can generate 194 kg of force (1903 newtons). With 170 cm cranks it's 324 newton meters. With 16T/38T gearing it's 136 newton meters at the hub. This exceeds Rohloff spec of 106 newton meters at the hub.
• Reliability is poor. There is no redundancy in the parts.
• Repairability with field tools is nonexistent.
• Hub gears are heavier.

Consider for example what could happen for my derailleur gears in a trip.

• Chain tensioner or rear derailleur breaks: I reroute the chain and make my bike a single speed.
• Front derailleur breaks: I use only one chainring
• One chainring breaks: I use the other
• One sprocket breaks: I use the other sprockets
• Chain breaks: I make it shorter and avoid the forbidden gear combinations

There is no conceivable failure of a derailleur gear system that would need me to walk home apart from perhaps freewheel failure, and this failure mode affects internally geared hubs equally well.

Some supposed advantages and one true advantage:

• Chain cases: nope. I have a chain case in my derailleur geared touring bike, so chain cases are not exclusive to hub gears.
• Coaster brake compatibility: not useful. The coaster brake is an on-off brake and it's on the wrong wheel so it's of no value.
• Stronger rear wheel due to more equal spoke tension: nope. Many modern bicycles are built with an asymmetric frame, resulting in more equal spoke tension even with derailleur gears. Besides, with proper wheelbuilding skills, good rims, good triple-butted spokes, 36-spoked hubs, etc, wheels with inequal spoke tension can be strong.
• Fewer chain changes: true.

In my opinion, hub gears are valuable only if you absolutely hate chain changes. However, then you must hate chain lubrication as well and it's a problem affecting hub gears too.

I've got shimano internal hub for 5 years mainly to commute.

pros:

• no maintenance - at the beggining
• easy changing at red light or to jump on the sidewalk

cons:

• harder to change tire when flat
• one day a car bump my wheel: I add to change the whole wheel+internal hub
• after some time (3000km) some gears are screwed, so it "jumps" when I press too strongly
• there's only one cogs, so it becomes old faster
• the lacks of derailleur force you to tune the chain tension very often
• or when you chain jump, it's harder to put it back

For all these reasons, I switch back to a classic XYZ-speed bike.

I drive the same Rohloff for over 8 years and do not see an end of its lifetime. I cannot speak for other brands, but in all posts above the free-of-charge service by Rohloff, as part of their special corporate culture was not mentioned. Up to now, whenever there was a problem, the thing was being send to Rohloff and I had to pay nothing. That being said, yes there are some problems worth mentioning and I assume these can apply to other internal gears as well:

• Hard to impossible to shift under load, which is the major reason why I have an additional bike with external gears that I use when going to the Alps.
• At tempeatures below 0 or -10 °C, the gears loose grip, i.e. you turn several times without moving. Up to now, after some turns, it started to work again but I always fear that it will stop working at -20 °C in the middle of nowhere. Rohloff recommends to use a special oil mixture during winter time which does not prevent this.
• Even in summer time it rarely but does happen that my Rohloff shows the slipping effect mentioned.
• You should have a LBD who knows Rohloff (including how this company is operating), which might not be easy to find. Moving, I had to turn my new LBDs into experts for two times.

The main advantages of hub gears over a 1xN derailleur system are reliability and being able to shift while stationary. Since there's no derailleur or chain tensioner it's easier to add a full chaincase which means chains last a lot longer and with mudguards added there are no dirty bits accessible on the bike. They're heavier and harder to maintain if you have to maintain them.

There's a whole style of bike that uses those advantages (and others) and from a US perspective is called a "european" or "dutch" bike. Broadly, it's a bike built for someone who is not a cyclist, rather they are someone who uses a bike for transport. It should just work, no special clothes or rituals required.

Compared to a 3xN derailleur system a hub gear is much simpler to operate and maintain. All the gears are in order and you only have one shifter. The compromise is a smaller gear range (Rohloff being the obvious exception), ranging from a little smaller for the Shimano Alfine 11 down to "yes, there are two gears" for the back-pedal-shift 2 speed hubs. But in return you get extremely long life for the chain and cogs (10,000km or more for the chain, 20,000km+ for cogs) and a general low-fuss maintenance requirement. That also makes it practical to put a hub gear in places where derailleur gears could go but you wouldn't want them - under my load carrying quad, for instance, where the derailleur hung down between the rear wheels just waiting for a rock to break it off, and turning the thing over for maintenance is a pain.

Removing the rear wheel on a hub geared bike ranges from nigh on impossible (some of the Shimano hubs, and some dutch bikes have chain cases that are just irritating to work on) to almost trivial (a Rohloff with QR). The problem is the shifter cable - you have to detach it to remove the wheel. Good ones just unhook, poor ones you have to undo the adjuster barrel and re-adjust the gears on reassembly. Shimano are getting better, but Rohloff and Sturmey Archer have it down pat.

Servicing hubs is not often done these days. The manufacturers generally allow bike shops to swap out the internals for new, saving the hassle of building a new wheel, but that's about it. I had three Shimano Nexus 8 hubs fail at 5,000km intervals, and every time I just got replacement internals rather than the hub being repaired (the last time, of course, Shimano had stopped making the hub so I had to throw away the wheel). With Rohloff the same thing might happen, except after 150,000km or more... most people don't ride that far in a lifetime. Also, with Rohloff you change the oil every 5,000-10,000km, at a cost of half an hour of your time and about $20 for an oil change kit (or $10 for just the oil).

I usually run a Schwalbe Marathon Plus on the rear of my hub geared bikes, which trades a little extra rolling resistance for huge puncture resistance. It's part of the "my bike just works" philosophy. I'd rather reliably take an extra 30s to get to work than at random intervals be 5-10 minutes late and dirty due to a puncture (because you know that you'll only get punctures when you're in a hurry and it's raining).

Pretty much as above. They are generally dependable, maintenance free, and easy to use.

Also heavy, difficult-to-impossible to service, and sometimes a bear to get loose if you have a flat tire. I fixed a flat on a Nexus hub wheel a couple of years ago, and getting the thing back in place and getting the cable properly lined up and tensioned was...Frustrating. I understand the new ones are better.

In fact, this one was so difficult that the previous owner had left a flat tube in the bike and had inserted one of those "straight" tubes with the sealed ends... On top of the old tube. Really strange....

I don't have an internal hub, but I want one for the following reasons: They are sealed and protected from the elements. They are nice for commuting because you can shift them while stopped... If you've ever stopped at a red light on a normal bike and struggled to get going again because of your gear, you can appreciate this. With an Internal hub, you can just change to an easy gear and then get going.
Also, I've had a derailleur brake off before and I've had a different one shift into my spokes! These failures will not happen with internal hubs.

Disadvantages are that they add a bit of weight and tie you to that wheel. That is, you can't just switch out wheels like you can with a derailleur. Finally, while internal hubs are seen as very dependable, if something does go wrong internally, you have to bring it to an expert. Wikipedia has a good summary: http://en.wikipedia.org/wiki/Hub_gear