As @Joe says, in most such contexts there's no discernible difference in meaning. But in some contexts there is a possible distinction between infinitive/gerund (or simple/continuous tense). For example,...
Always tell the truth, but don't always be telling the truth.
...which could be mirrored by something closer to OP's example, such as...
I want you always to tell the truth, but not always to be telling the truth.
In such constructions we're forced to acknowledge that the -ing versions emphasise continuously doing something (in this case, the intended sense is that whenever you do say something it should be true, but sometimes it would be better not to say anything at all).
But in OP's example it would be stretching a point to suggest that the first version means if she happens to be aware that people are looking at her, she likes it, and that the second version carries more the implication that she's only really content when people are looking at her.
It's also worth noting what happens if we substitute a different verb...
1: She wants to be looked at.
2: ?? She wants being looked at.
3: She wants looking at.
...where #2 wouldn't normally be considered grammatically acceptable, and #3 would almost always be interpreted as an idiomatic format meaning [objectively speaking] it would be better if she were looked at. Some people would allow that interpretation for #1, but it would more naturally be seen as equivalent (or at least, extremely similar) to both OP's examples.
Being a teacher, she likes children.
When I read this, I assume:
1) The woman is a teacher. She teaches for a living.
2) She likes children.
3) There is some relationship between her love of children and her profession. The exact nature of the causality is unclear – perhaps she got into teaching because she likes being around children, or perhaps she's grown to enjoy being around children because of her profession. Or maybe the writer assumes that all teachers like children. Regardless of the particulars, though, which are left unstated, the sentence seems to imply that the two facts are somehow linked.
Having been a teacher, she likes children.
This tells me:
1) The woman was a teacher. She used to teach for a living.
2) She likes children.
3) There is some relationship between her love of children and her former profession. Again, the exact nature of the causality is unclear – perhaps she likes children because they bring back memories of her time in the classroom. Nevertheless, this sentence also seems to imply that the two facts are somehow linked.
Best Answer
The answer in terms of grammar is that the first is present simple, and the second is present progressive. With an adjective as the object of to be, this would mean that "she is busy" is describing either her current state, or her habitual state, while "she is being busy" suggests that she is doing something that causes the adjective busy to apply.
However, the present progressive of to be, especially with an adjective, often requires some special handling. It depends on the adjective. After all, what can cause an adjective to apply? For example:
That means she is doing something that makes her naughty - or put another way, that she is doing something naughty.
This again suggests doing something that indicates cleverness, but if you just wanted to say she was clever, you wouldn't say that. This suggests that she's doing something for the purpose of being or seeming clever. You'll hear this about children talking back to adults - if what they say is deliberately misinterpreting the adult's words, or pointing out a contradiction, or so on.
This one is particularly complex. It might mean someone trying to be funny, even if they're failing. It might mean someone acting strangely, as that's another meaning of funny. It might just mean that they are being genuinely amusing, even if they don't mean to be.
This one is also complex. It might mean she is making a show of being busy, but it might also be a tart response to the question "what is she doing?".
If you just want to say someone is busy, the safest bet is "she is busy".