Having had a quick look at your link, they are not all fast growing trees - those that are have a growth rate per year listed. They appear also to be selling some 'mature' large plants - the drawback with those is, the older they are, the less easily they transplant into the ground compared with younger specimens.
And yes, there is a major drawback with fast growing plants, which is they may quickly outgrow the spot you've planted them in. Always check the ultimate height and spread of any tree or plant described as 'fast growing', and make your decision based on that rather than the overall speed of growth. Finding that information on the site you've linked to might be impossible though; in terms of using botanical names and height and spread, I couldn't find anything useful at all.