I agree with Sam, and would like to expand on his second point.
When we buy from an online agency (cheaptickets, Travelocity, Expedia, etc), through Disney Travel Company, credit card portals, etc they are our acting
travel agent for this transaction, just like if you went to your neighborhood
travel agency in the strip mall next to the pizza place.
While there are many advantages to using a travel agent, the main disadvantage is you basically relinquish your rights to interact with the airline, and grant those rights to your agent. If there are problems (weather, mechanical issues, canceled flights, need to end the vacation early, etc), then you have abdicated your ability to deal with the airline yourself. Thus, you would need the get the agent to make the changes for you.
While is is great for the local professional in the aforementioned strip mall and likely ok for Disney or a credit card's agency, problems arise using the discounted websites. Expedia, cheaptickets, etc are not really concerned about follow-up service. Their money is made with the initial transaction, and the volume they generate makes the threat of leaving them for future transaction mute. In essence, they don't car and won't work to help you in a difficult situation. In addition, they form relationships with the airlines and hotels (some good and some bad), so I would say they value those relationships over one single customer/transaction.
That said if you want to have control over your reservation, then book with the airline, hotel, and car rental agency yourself. You not only have recourse as a direct customer, but you also can apply frequent flier programs. If you would rather not worry about the transaction, use a reputable agent in your neighborhood, who will need to look you in the eye when you get back. Also, major travel agencies with good reputations (like American Express or
Dreams Unlimited) are also a good bet, since they are experienced and the travel section is a core business with them.
Long winded, but important for others on the board with the same concerns.
Personally, I use ITAMatrix and Bing Travel's predictor tool to find flights and get an idea if now is a good time to buy. Then I book straight with the airline (the Matrix tool provides all the info you'd need should a call to the airline's agent be required). I also book directly with hotels and car companies.
Good luck!