Three piece bows have great flexibility. They pack down smaller than other bows. You can have extra limbs made for them. With many three piece models, the limbs are made to be exchanged among different risers without needing to have the bowyer custom fit them. If you like additional weight for stability, the three piece bows are generally heavier. There are lots of choices of woods and ways to glue up the woods in the riser and limbs. Often you can change the lengths of the risers and the limbs to get the bow that fits you or change weights to accommodate an injury or different species of animals that you may be hunting. On a hunt to a distant destination you may want to have a spare set of limbs in case of damage or a lighter and heavier set of limbs for different animals. A three piece bow gives you that option. A very good example of the kind of bow I am talking about is a Morrison where you can choose riser materials, riser length, limb materials, limb length, limb design, draw weight, and then mix and match them if you want to. Other excellent bowyers offer similar products. If you want a very versatile design, go with a three piece, especially one with interchangeable limbs and risers.
I am also very fond of two piece bows, especially those with Bow Bolts. As stated above, most good two piece bows are basically a one piece bow that has been cut in half with a stable connection between the two halves being added during the construction process. A good connection, like one created with a Bow Bolt, is almost unnoticeable until you want to take your bow apart. Aside from a slight weight increase and a thin area in the grip where you can see the joinder of the two halves, the bow looks and shoots like a one piece. The two piece bows tend to be much lighter, slimmer and handier than three piece bows, once they are connected. The two piece does not break down into as small a package to transport, but if you pick a fairly short bow like a Shrew Classic Hunter or a Shrew Lil Favorite (wonderful bows!!!), they are still very short. Even a 62" two piece like my superb Holm-Made Osprey can be broken down, stored in a case and put into a longish duffel bag. The disadvantage of a two piece is that there is no interchangeability among the constituent pieces. What you see is what you get. Two bows of the same make and model are totally separate, and never the twain shall meet, mix and match. They are definitely not as versatile as three piece bows, but they are much nicer to carry around all day due to their light, handy design. If you prefer a one piece bow but want to travel with it, get a two piece. There are connectors other than the Bow Bolt, but I believe it creates a far superior connection if you want the ultimate in stability, rigidity, and strength. However, the bow must be designed and constructed with the Bow Bolt in mind to achieve its potential. This is not an aftermarket solution.
Most of the time my bows are put together, not taken apart, but when the time comes to travel with them, especially by air, take down is essential. Why limit yourself to either a two piece or a three piece? Buy both!
Allan