So lets start with the Catamarans:
Speed - Speed always comes up in my research as one of the advantages of a catamaran over a monohull. Although this is the speed of passage making which appears to be 20% faster than that of a monohull.
Stability - Catamarans have minimum heeling which makes them a very comfortable ride and a stable platform while anchored out. This means that you can serve dinner and drinks on the table and not worry about having a mess on the floor.
Privacy - Although I've been inside monohulls that can provide some privacy between 2 rooms if the boat is big enough, it can't even compare to a catamaran. Catamarans have two hulls separated by a platform in the middle which provides unmatched privacy.
Living Space - A catamaran has more living space compared to a monohull. You can hang out on a large deck or inside the cockpit. You don't have to crawl in through a ladder every time you want to enter the boat. There is a 360 degree view from inside the cockpit and usually lots of ventilation. Catamarans were made for the tropics.
Accessibility - With a catamaran you can have access to some of the most beautiful and secluded shallow waters in the world. You can even beach a catamaran. This is due to the fact that catamarans have mini keels that are 4 feel deep or less. Also, you can have a catamaran without the mini keels and a retractable daggerboard instead. This would give you even closer access to some of the most beautiful beaches in the world which a monohull with its large keel could never get to without the risk of getting stuck.
Safety - Safety is also a heated debate between monohull and catamaran sailors. I think that there are certain advantages a catamaran has when it comes to safety that cannot be ignored. I've read stories of catamarans staying ahead of the storm and essentially outrunning it due to the 20% speed advantage over monohulls. It gave them the extra 2 knots speed required to stay ahead of the incoming storm. Catamarans are built of material that floats so they are virtually unsinkable. Personally I like the idea of a boat that can't sink. It ends up upside down but none the less you still have access to your food and supplies while you await rescue. Catamarans are made of separate watertight compartments which means that if a couple of those compartments were to fill with water you can still sail your cat back to a boat repair yard. Bottom line; a hole in a monohull would sink the boat in a few minutes at most. I hope you can get everything you need in those couple precious minutes to help you survive until rescue arrives.
Expensive - Catamarans are expensive sailboats. This could be a reason why the average live aboard will choose a monohull over a catamaran. You can find a true blue ocean going monohull that will take you around the world and provide you with the comfort to live aboard for considerably less money than a catamaran of similar capability. Some will argue that you have more living space on a catamaran and when you factor in all the space in square footage and compare it to a monohull the price difference is small. The point here is that you can find a 32 foot monohull with the safety and storage volume capacity to take you around the world, you cant say the same about a 32 foot catamaran. Therefore since you are looking at a 40 foot plus catamaran for a safe passage and sailing around the world you will be paying considerably more money for it.
Sailing Upwind - Catamarans cannot point as high as monohulls when sailing upwind. Those extra few degrees can make a big difference. Although, catamarans with daggerboards sailing upwind can get to their destination as fast or even faster than a monohull even when not pointing as high. Having said that overall monohulls point higher and have better performance upwind.
Docking - You are looking to pay more for docking space for a catamaran. Due to a wide beam you may even have trouble finding adequate space in a marina for your catamaran. You will require a larger space which equals more rent money per month. This is where you can utilize the advantage of a catamaran's stability and anchor so you can avoid paying at a marina as much as possible.
Load Carrying Capacity - This is very important when planning on sailing around the world. When you make a passage you require a lot of supplies. These can weigh down your catamaran and make it very sluggish and take away some of the advantages such as speed. In order to keep the performance advantage of a catamaran you have to utilize modern technology and use light weight equipment.
Affordable - Monohulls are a lot more affordable than catamarans when looking for a monohull that will do the same job as a catamaran: take you on an adventure sailing around the world. The resale value appears to be better for monohulls as well although some notable catamarans have a very good resale value especially in last few years given the rising popularity of catamarans.
Sailing Upwind - A monohull will point higher upwind and have better overall performance sailing upwind compared to an average catamaran. There may be some exceptions with dagger boards but overall monohulls have a better upwind performance.
Load Carrying Capacity - Given that a monohull has a very heavy keel and is a very heavy boat it has large load carrying capacity due to its strong and rugged construction. Its very heavy compared to a catamaran and therefore heavy supplies will not affect its performance as much because it was already designed to sail as a heavy boat.
Docking and Storage - Monohulls are cheaper to haul out for repairs and maintenance. A slip for a monohull is less expensive as well. Monohulls don't take up as much space as a catamaran because they dont have sa wide of a beam, therefore you have alot more options with a monohull when it comes to storage.
Traditional - Monohull sailboats have always had this romantic aura about them. For centuries there have been tales and legends of adventure at sea all involving monohulls. Some people just want to feel and experience the history behind the traditional monohull. It makes them feel like they are stepping into the shoes of the famous explorers throughout the centuries who set out to explore the oceans around the world. Having said that, catamarans have a centuries long history, especially with the Polynesians who sailed catamarans throughout the Pacific, mostly very small catamarans too long before the Europeans set out to explore the World.
Heavy Keel - Without the keel a monohull would not be able to stay upright. However, with a heavy enough keel to keep it upright come many disadvantages. The boat is built heavier and more rugged to support the keel and heavy rigging and therefore you get a slower boat. I can see the advantages of a heavy boat being more stable in the water but having a sluggish boat is a definite disadvantage. Another big issue is accessibility to some of the most beautiful shallow waters and beaches in the world. If you plan to sail around the world and to visit some of the most beautiful, romantic and secluded islands and atolls then you're going to have a problem with a huge chunk of weight underneath the boat preventing you from entering any of those atolls or approaching those pristine beaches and breathtaking islands.
Speed - Speed may not be a big issue for a live aboard since they are not planning on racing around the world, they just want to enjoy the journey and if it takes a while longer so what. If on the other hand you value your time then a 20% faster passage on a catamaran will give you a few extra days to spend however you please. Also, as I mentioned before , the extra 2 knots can keep you away from a storm.
Safety - I've covered the safety advantages of the catamaran earlier. Here I will just summarize that a hole in a monohull will make it sink in a couple minutes where as a catamaran is virtually unsinkable.
Privacy - There is very little privacy in a monohull and considering that you're going to be stuck with a couple people for many weeks to come, no privacy can be a big issue! The separate accommodations in each hull of a catamaran gives people a lot more privacy compared to a monohull.
Now, for the conclusion. I had mentioned at the beginning of this post that I have already made a decision about which kind of sailboat I would like to take sailing around the world. My choice is a Catamaran and it is due to the following reasons: Safety, Privacy, Stability, Living Space and Layout and Accessibility to the most secluded places in the world. I do have to admit that if I do end up changing my mind and going for a monohull it will be strictly due to affordability. If a monohull becomes my only option of sailing around the world due to the high cost of catamarans then I will stick to a monohull.