Here are some of the pitfalls:
You have to make an appointment to meet the "keyholder" and take possession of the apartment. They usually ask you to call them to confirm when you're leaving the airport, or when you arrive at the train station. This means you have to have a way of making the call. Be aware that many of the so-called public phone charge take only credit cards and charge outrageously high fees for local calls. A hotel doesn't require an appointment.
You usually have to pay a nonrefundable deposit to rent the apartment, and then pay the full balance in cash when you arrive, before they'll give you the key. Once you've handed over the cash, you have no recourse if things go wrong, so you have to do a very thorough inspection before paying. The person who's waiting to be paid usually doesn't like being kept waiting, of course. At most hotels, you pay when you check out.
Checkin time (at both hotels and apartments) is usually in the late morning or early afternoon. A hotel will allow you to drop off your bags when you arrive, so you can go our to sightsee without dragging your luggage around. With an apartment, you either have to keep the luggage until you can check in, or else go to the train station to check it for a few hours and then go back to retrieve it.
A lot of the agents handling apartments are really just listing services; the owners are responsible for the condition of the apartment, and the agency will often wash its hands of responsibility if there is a serious problem. Some agents have layers and layers of businesses and it's very difficult to pin the blame on anyone, and next to impossible to deal with the owner.
Many apartments are listed with multiple agencies; this can lead to double or triple booking, and the famous "Apartment not available because of water damage" notification that may arrive just days before your departure. You have to be very, very careful in choosing an apartment and look for reviews on multiple web sites, both of the apartment and the agency. Slowtrav.com is a good site for checking reviews, but most apartments haven't been reviewed at all. There is a huge turnover in apartments.
You won't have a front desk to help you out if you need information or advice, or want someone to call a taxi for you.
The vast majority of apartments is rented under the table, which means there are have been no safety inspections. I've seen very serious safety problems in apartments I've rented.
Edited: 4:46 pm, November 01, 2012