Advantages of a Trailer Over a Motorhome
Price. Trailers don't have the expense of an engine, and so cost much less than equivalent sized motorhomes. If you add together the price of a new tow vehicle and a new trailer you may well get to the same price point as a new motorhome, but there's an important thing to keep in mind: almost all RVs have very limited warranties, often only 1 year. If you end up with a lemon of a trailer, you'll be out a lot of money, but it will be a whole lot less than if you end up with a lemon of a motorhome.
Day-tripping options. With a trailer, you can always unhitch and have a (relatively) normal sized vehicle to explore with. With a motorhome, you will need to tow a car (or motorcycles or something) if you want to be able to venture out and explore without breaking camp and having to drive your large rig to visit the sites.
Ownership costs. If you currently own a vehicle capable of towing a trailer, then you don't have another motor vehicle to service, maintain and insure. If a motorhome is going to sit unused for a long stretch, it will likely require more maintenance than a vehicle that gets driven regularly. Insurance on a trailer is also much lower than on a motorhome because it's not a motor vehicle. If you do end up buying a second vehicle for towing, it can at least also serve as a secondary vehicle, so can be useful even when you're not travelling.
Transporting a family. Motorhomes may advertise that they can sleep more than 2 people, but they are seldom if ever made to safely transport more than two. Almost all seating past the first two bucket seats are only cushions on top of plywood boxes, which is not a truly safe transport option when barrelling down the highway at 65 miles an hour. Car seats for young children in particular will have no where safe to latch in. Purchasing a tow vehicle that seats 5 or more safely, on the other hand, is pretty easy.
Selection (espically in the used market). If you go to any RV show, you'll see dozens upon dozens of travel trailers, but generally very few motorhomes. You'll find a similar situation at typical RV dealers, except those that specialize in motorhomes. And if you're in the market for a used RV, you'll see more than 4 times as many trailers and 5th wheels on RVTrader.com than you will motorhomes. So if you want to have lots of selection when it comes to layout and price point, trailers win hands-down.
Advantages of a Motorhome Over a Travel Trailer
No towing required. This is probably the number one thing that puts people off of trailers and leads them to buy a motorhome. Towing can be intimidating even for the most confident driver, especially when it comes to reversing a large trailer. There's also a huge learning curve when it comes to matching tow vehicles to trailers, and buying a motorhome means you don't have to worry about any of that. You also aren't limited in what you can buy based on what your vehicle is capable of towing. Of course, if you plan to tow a car for day-tripping, then towing may still be in the cards for a motorhome, but those are almost always unhooked before you arrive at a site and need to back in, so that reversing while towing isn't an issue.
Ease of setup and teardown. Motorhomes have a huge advantage here over trailers. For most motorhomes, setup pretty much consists of throwing down a couple of levelling blocks, and putting out the slide if you have one. For trailers on the other hand, there's the extra step of unhitching, which sounds quick but takes a good amount of time, especially when weight distributing hitches and sway bars come into play as they add extra steps to the process. The same is obviously the case for breaking camp, and if you're ever looking to make a quick escape, you're not likely to be doing it in a trailer.
No surprises when you open the door after a trip. The moment when a trailer owner first opens the door after arriving at their destination is one of ultimate suspense - what happened during the drive? Did the fridge come unlatched and will food be all over the floor? Did the anchors holding down the microwave finally come free and was it launched across the trailer? In a motorhome, usually somebody notices the second a cabinet opens up and you can pull over and rectify the situation before it gets worse. (Of course, this barrier could be a plus if you don't want to hear the rattle of the pots and pans while you're driving!)
Stealth camping. In a smaller motorhome you can often park on the street or in a parking lot and then retreat to the rear where you can quietly live with the blinds closed and nobody will be the wiser that you're not just parked there and staying in a nearby home. With a trailer, at the very least you need to get out of your tow vehicle and get into your trailer, which may be enough to give you away.
Being off-grid for long periods. Most (although not all) motorhomes are built with RV house batteries that will charge off of the engine's alternator while driving. So if you've spent the night boondocking in one location and are travelling to another boondocking location, your batteries can recharge while you drive between the two. With a tow vehicle and trailer though, by default the tow vehicle's alternator will reduce its output voltage once the truck batteries are charged, so the RV batteries will not charge much after that. While it is possible to arrange it so that your tow vehicle's alternator continues to charge the trailer's batteries while driving, some fancy electrical work will likely be necessary to make it so.
There's an RV out there that's right for you, hopefully this list of pros and cons will help you get the ball started rolling in the right direction! If you have any big advantages or disadvantages one way or the other that I've missed, please add them in the comments!