It seems like you could a second bucket for the secondary fermentation. Is there any drawback to not using a carboy for this purpose?
You want to reduce/eliminate oxygen in your beer once it's past fermentation.
Plastic buckets let in an extraordinary amount of oxygen, over time, so you should avoid them if you can.
There's more information here, https://homebrew.stackexchange.com/questions/68/whats-the-point-of-secondary-fermentation, and elsewhere in the site.
Just as a note, I never transfer to secondary, and many others here believe the same. I don't think it gives any benefit to most beers.
You want less head space on your secondary to reduce the amount of oxygen that comes in contact with your beer. Once the initial fermentation is done in your bucket there is a nice thick layer of CO2 that sits in the 1.5 gallons of head space in your 6.5 gallon bucket. But, once you move it to your secondary the fermentation process is already done, so your beer won't make another thick CO2 layer to protect itself. Therefore you want to put a 5 gallon batch into a 5 gallon carboy to eliminate that head space while it clears itself up in your secondary. You want to eliminate any chance of oxidation at this point before you bottle and enjoy.
If all you can afford or if all a brewer has is a second bucket it will be fine for doing a secondary. You do want to use a smaller bucket if possible to minimize head space if you are using it for storage while aging or clarifying.
If you are doing a true second ferment due to the addition of fruit or sugary syrups, then a plastic bucket is indeed the right choice as there will be some fresh fermentation. And its easier to clean that out later especially if fruit was used.
If the beer is done fermenting there really is no reason to do any secondary at all. Just let the beer sit in primary for at least 14 days in general and you'll get beer just as clear as if you used a "secondary" (plastic or glass). Without the hassle of another step, more cleaning and sanitizing and less oxygen pickup in the final product.
I believe that along with the other answers here, many times people simply trust Glass to be less likely to leach undesirables into their beer. This may be a matter of superstition more than science, but it is a fact that plastic is porous where glass is not. Also your typical Carboy and airlock (or just a simple stopper) leaves less chance for nasties to enter your precious beer than a bucket lid might.
As long as you don't oxidize your beer while racking, transferring to another plastic bucket shouldn't have a negative impact on your beer. The lingering C02 blanket concept is a myth.
C02 will seek equilibrium and dissipate in a plastic bucket primary fermenter. It is just as much susceptible to oxidation there after primary is complete as it would be in a "secondary" plastic bucket.
This is not to say that it isn't worth it to use a carboy with a narrow neck. Here, over time, less of the surface of the beer will be in direct contact with oxygen. You might be able to get a tighter seal resulting in less air flow. You can see what's happening to the beer through the clear container. It's easier to pop open and add adjuncts, or pull a sample.