I will be watching TV.
I am going to watch TV.
To me, they mean the same thing? Do they in informal speech? What about formal speech?
There's also "I will watch TV".
All the different future tenses that show up in English have differences. Sometimes they are major, like the future perfect, and sometimes they are subtle. Sometimes the difference in meaning depends on context, or on even on what the verb is.
"What will you be doing this evening?"
"What are you going to do this evening?"
Those questions seem basically the same, don't they? They are the corresponding questions to your two examples. Either answer could be given to either question. But what about...
"What will you be doing at 9pm this evening?"
Suddenly the fact this is in the progressive aspect matters, because it's at a precise time. The more precise the time, the more the progressive (or lack thereof) matters. This question means "what will you be in the process of doing at 9pm". Contrast
"What are you going to do at 9pm this evening?"
That isn't in the progressive. It suggests there is some thing you are going to either do in a very short time at 9pm, or start doing at 9pm. It would be a weird question to ask unless you were trying to make sure someone understood a plan or something.
In the first "9pm" question, your answer should also be in the progressive, and in the second it should not. However, without that context, and with the example of "watch TV", they are pretty much interchangeable regardless of formality. On the other hand, if the activity weren't something that one tends to do on an ongoing basis, the difference might be significant. Consider:
"I'm going to push the button."
"I will be pushing the button."
The first says you're going to push the button. The second says you're going to spend some unspecified but non-trivial period of time pushing the button (though context can change that).
Finally, it's important to note that it's the progressive aspect that makes the major difference between the two. You could also say:
"I will watch TV."
"I am going to be watching TV."
These differ from the earlier examples largely in nuance. The difference in meaning between the two is still almost entirely from the fact one is progressive, and the other is not.