Java is a Sun product that is closed source. Recently, they were interested in releasing Java under an open source license. However, they couldn't release it that way because small portions of java had already been licensed to other companies.
Instead, they took the part that belonged to them, made it open source, and have been trying to fill the gaps since by working backwards. However, this has not made a perfect final product. Therefore, they had to release this as a separate testing project called Iced Tea. There is a similar situation with the open source drivers made by the community for hardware from companies such as ATI and Nvidia. The community works backwards and tries to do the best they can, but the product is not always stable or completely functional.
So what are the disadvantages of using Iced Tea? For the casual Java user, not much. Some of the parts that were missing, like the font renderer, may slightly decrease the user experience, but there shouldn't be major issues. However, some advanced Java functions may not work. If you haven't experienced any issues so far, I would say you would be fine sticking with Iced Tea. The only thing I could think of is a possible performance hit, but I haven't heard of this happening before.
Just a question: Why use IcedTea? If you especially wanted to use open source alternatives, you probably don't want to use Linux Mint (other than the light edition). However, there is one advantage of IcedTea: in countries with strict software patents (like the US, Japan, etc...) open source alternatives like IcedTea are legal in open source distros, unlike their proprietary counterparts. Also, if you were interested in investigating more in open source alternatives, you can install the Gnash flash player (search it in synaptic). This replaces Flash, but may not fulfill your Flash needs. Currently, there aren't really open source handlers for commercial DVDs, mp3s, wma, etc... although you could support open source by using their alternative completely open source formats like .ogg and Theora video. Or, you could use more mainstream formats that are closed source, but not as locked down, such as mp4, avi, etc...