It depends on personal tastes (both aesthetic and flavour).
Ceramic pots certainly retain heat longer which suits black tea as a pot can be on the go for a while.
Clay is for the connoisseur and should be kept for one specific tea.
These ideally need to be used regularly so don't suit most households (including me).
I'd personally have: A western style ceramic pot (or two, one small, one large) for black teas like Kenyan,Ceylon or Assam based teas which I'd only rinse out (unless it got forgotten and grew mold!
in which case it gets a good scrub) A glass pot for strongly flavoured teas or heavy tannin tea like Lapsang Souchongor chai (if i make it in a pot) which would be cleaned each time
An eastern style ceramic pot for green tea as they generally keep the larger leaves back.
The biggest difference I've found is that warming the pot is more significant when using ceramic or clay over glass as the material absorbs more heat.
cooking.stackexchange.com/users/2391/vwiggins
Clay is porous.
The aromatic elements of a tea will stay inside the pores and come out for the next brewings.
After a few brewings, the teapot will be "coated" and the aromatic peaks of the particular tea will be emphasized, giving it a more satisfying taste.
For this reason, clay teapots should be reserved for one family of tea only (e.g. flowery oolongs, pu-erh, wuxi rocks tea, high mountain taiwanese, etc.).
It would make no sense to enhance the particular flavors specific to leaves from a category A in a brewing of tea of a category B with different aromas.
Yes, that means that if you like different families, you must invest in one clay teapot for each.
Glass, iron or ceramic teapots do not show this behavior and can be used with any category of tea during their life.
Also, clay teapots have been traditionnaly used only for oolong and pu-erh teas, and some Chinese black ones.
It's quite unusual to brew green, white or Indian black teas in clay teapots.
It may enhance those kinds of tea too, but you would have to make your tries by yourself to estimate that.
As a side note, since clay keeps flavors, never wash a clay teapot (or cup) with detergents !
Plain hot water will suffice.
cooking.stackexchange.com/users/3283/Manur
I find with glass teapots they get extremely hot, where a ceramic tea pot does not.
In my experience the Tea also stays hotter in a ceramic teapot.
The ceramic teapots are harder to clean though, and you can't see the tea brewing as easy.
cooking.stackexchange.com/users/1675/Manako
OK.. I guess there is no right or wrong answer for this question, so let me start this with a story.
"Many years ago, there was a poor guy in a remote Chinese Village and he owned a clay teapot and the teapot had been using for generations.
One day a city man came to the village and saw the teapot
and he was very impressed by the teapot, so he decided to buy the teapot and paid thousands for it.
He told the village guy that he would pick it up next day.
The village was so excited and he reckoned he needed to give the pot a big clean, so the buyer would be happy about it.
Next day, the city man came back and saw the teapot was very clean, it was so clean that there was not stain at all.
The city man then was very disappointed and decided to drop the sale.
The village man didn't know what went wrong and just found out that the teapot wasn't worth anything, but the tea stain that got built up by generations was worth the most."
This story tells us that Chinese have a long history of using clay pots and the clay pots themselves enhance the tea favour because the "stain" adds favour to tea.
Also, temperature can be controled better with clay pot.
REMEMBER the mort IMPORTANT THING ABOUT TEA!!!
You are not supposed to leave tea in the teapot for more than couple of mins as the acids from tea would come out if you leave the tea in pot for too long.
What I wanna say is advantage of using clay pot will enhance tea favour, but it's a very personal choice.
Glass pot is fine and just don't leav tea in the pot for too long.
Also, avoid using metal pots as the chemical reactions between tea and metal don't go too well.. Cheers
cooking.stackexchange.com/users/2951/Foodrules 
I think that a lot of the difference is aesthetic.
Tea will eventually stain anything, including glass.
If the stains are inside the creamic teapot, they won't bother anyone, and can quietly add a touch to the flavour.
In a glass teapot, the stains are visible from outside, and might put some people off their tea.
cooking.stackexchange.com/users/611/Carmi
Won't break if you drop it?
Other than that, you're bound to get a metallic taste out of any metal teapot (that isn't enameled, and most of them will be) that you use to make any tea with even a hint of acidity (most teas are somewhat acidic).
The more reactive the metal (copper, silver, iron) the more taste.
Of course, the tea cools off super fast in those as well, so it doesn't have much leach time.
I'd stick with glass for the best experience, though I've seen a number of restaurants use small stainless steel pots to good effect.
cooking.stackexchange.com/users/218/Satanicpuppy
Glass and ceramic teapot are pretty similar except that the glass teapot will get really hot, but you can see your mix of tea in it
and it's easier to clean.
Have a nice cup of tea Ryan!
cooking.stackexchange.com/users/2877/Nick
I currently have a borosilicate glass teapot.  Glass seems to have the advantage that it will not absorb flavours and so can be used for all styles of tea.    What are some of the advantages of using a clay or ceramic teapot over glass?  I am not interested in the differences in style or size, just the material used.  Edit: anyone with information on clay teapots, such as the Yixing, and its (dis)advantages?
Advantages of using a non-glass teapot?
cooking.stackexchange.com
<equipment><tea>
cooking.stackexchange.com/questions/8363
cooking.stackexchange.com/users/2912/Ryan Anderson
Clay teapots retain heat longer and don't get as hot on the outside, but they are porous and easily get stained or retain the flavours of tea used. For this reason, glass can be used with different kinds of tea, but different clay teapots should be used for different families of tea. Glass teapots also show the contents, while other materials do not. Clay should be kept for one kind of tea as it absorbs odours. Ceramic will retain heat for longer but are harder to clean. Glass pots will eventually become stained.
Ceramic will retain heat for longer. Clay is porous and should be kept for one specific tea. Glass teapots get very hot but they can be used for any type of tea. The same with ceramic. Ceramic teapots are harder to clean and the tea will eventually stain glass.