Hi Lola...
As a bike shop owner, we carry both, but we highly recommend the child trailers for numerous reasons... (I apologize in advance if this is a novel, LOL.)
Your child is very securely buckled in. Good models have 3- or 5-point padded harnesses much like a good stroller, so when buckled properly your child will not crawl out.
The trailer is MUCH safer. Should the parent hit a rock, loose their balance, or have any type of mishap, if the bike falls the trailer stays upright and does not tip over due to the swivel on the hitch. Because of this, the child also does not cause your bike to shift or sway if they shift their weight or move around like a seat does.
Also with a quick-release style hitch, you can easily remove the trailer if you want to go somewhere without it. The quick-release is sometimes an accessory, so be sure to look for it. Much easier attachment and removal. Some models also have a stroller attachment to convert it to a pushable stroller. Some people like this. I don't use the feature on ours as I feel it is too bulky as a stroller and I love the stroller I own. But it is an available option on some models.
The trailer can be used for larger children so it will last you longer. Even though our oldest son loves to ride his bike, if we go on a long trip or are going into town when we're camping he rides in the trailer with his little brother. (We own a Burley D-lite double).
The trailer is enclosed giving you lots of nice features... The screen keeps bugs, leaves, etc. off of the child but allows the breeze in. Ours has an adjustable sun shield to block the rays, and also has a clear vinyl zip cover so should it rain or get too windy or is a little chilly, the kids are protected but can still see where they're going.
The kids can easily have a book, toy, sippy cup, etc in the trailer with them. Obviously you can't do this with a child seat.
There is room in the trailer behind the seat to store things like coats, change of clothes, diapers/pull-ups, snacks, whatever...
Good trailers can be pricey, but worth the investment. This is especially true if you plan to have another child. If you plan to have another soon, then consider a double - more room for your daughter now, and for both children later. Otherwise, the single is great. Good trailers also tend to hold their value for resale when they've been well taken care of.
If you do choose a child seat, look for one with a recline feature. Helps keep a sleeping child a bit more comfortable.
Hope this helps you in your decision. If you have any questions in your search, I'd be happy to try to answer them for you - ____@____.com luck!
~ V.