When it was just my husband and I, hotels made more sense. With a larger group (i.e., our kids, in-laws, parents/siblings, etc.), apartments are definitely the way to go for us. Also, if you have smaller kids, then an apartment with a kitchen and washer and a separate bedroom is nice to have. Also, if you have an extended stay (such as a week), then perhaps an apartment would be more comfy.
As others have noted, really look over the apartment's written details, reviews, and location on the map. Email the owner a benign question to get a sense of them. I have rented apartments almost exclusively for the last few years through mostly AirBnb in the States and other companies for abroad, and cannot have been more pleased.
The only drawback I have come across (especially abroad in a big city like Rome) is the fact that you need to call them to meet you at the location. I have not had a working phone in the past and have had to rely on the kindness of strangers (usually a group of folks sitting at a nearby cafe). I do not tend to ask the front desk folks at hotels questions or directions, so not having someone there has not been as big an issue. Another perhaps con for an apartment rental is that if you sometimes put down a deposit, and then have to pay the remainder in cash as soon as you check in, so if the apartment is your first stop in the trip, then you may need to do some exchanging of money prior to the trip, so check to see how you need to pay for the place (not an issue with AirBnb since you pay for the whole thing up front but more for some other companies that have this arrangement).
Either way, it's always been a fun time!