Learn English – the difference between “would visit” and “would be visiting?”

As other answers have indicated, the future progressive, as the term implies, can place emphasis on the ongoing nature of the future event. In such contexts the future simple is not correct:

This time tomorrow I'll be lying (*I'll lie) on the beach.

Sorry, I won't be able to make it. I'll be playing (*I'll play) tennis with Mike.

However, this is not the only use of the future progressive. It is often used when there is no particular focus on the ongoing nature of the future event:

You'll be hearing from my lawyer.

She'll be starting school soon, won't she?

In such cases, the future simple is also possible:

You'll hear from my lawyer.

She'll start school soon, won't she?

although, to my ears at least, these are very slightly less natural.

Now we come to the OP's examples, both of which are perfectly normal ways to tell someone of your friend's holiday plans.

There is possibly one small semantic difference, however. Namely, that the progressive form could carry with it the implication that the visit is part of an arrangement, whereas the future simple is a simple statement of fact. As such the future progressive parallels the use of the present progressive to express arranged future events:

I'm playing tennis with Mike tomorrow.

I'm visiting my grandparents at the weekend.

She's visiting Chichen Itza next week.

But in Fumblefinger's term, this is "armchair rationalisation", a process that no native speaker consciously goes through in advance of what they say in day-to-day conversation.

In summary, the OP's two sentences are virtually equivalent, but there are other contexts where only the future continuous is possible, or where it may sound a little more natural than the future simple.

A very thoughtful and hard question indeed; it pushed me to research a bit on the subject Continuous Conditional.

Your first example is a past perfect continuous sentence. In general, it is used to indicate an action was happening before another action happened. However, it can also be used to indicate past unreal condition. For example:

If I had been talking to him when he said that, I would have punched him in the face.

But fortunately, he was not talking to him when he said that and that's how he missed my punch.

So according to this theory, your first sentence which is

If I had been with you, I might be taking care.

is absolutely meaningful and grammatical.

Now, there is a vital fact when using these type of constructs as described by data.grammarbook.com,

When talking about something that didn’t happen in the past, many English speakers use the conditional perfect (if I would have done) when they should be using the past perfect (if I had done).

For example, you find out that your brother saw a movie yesterday. You would have liked to see it too, but you hadn’t known he was going. To express this, you can use an if – then clause. The correct way to say this is with the past perfect in the “if” clause, and the conditional perfect in the “then” clause:

Correct: If I had known that you were going to the movies, [then] I would have gone too.

The conditional perfect can only go in the “then” clause — it is grammatically incorrect to use the conditional perfect in the “if” clause:

Incorrect: If I would have known that you were going to the movies, I would have gone too.

More examples:

Correct: If I had gotten paid, we could have traveled together.

Incorrect: If I would have gotten paid, we could have traveled together.

Correct: If you had asked me, I could have helped you.

Incorrect: If you would have asked me, I could have helped you.

The same mistake occurs with the verb “wish.” You can’t use the conditional perfect when wishing something had happened; you again need the past perfect.

Correct: I wish I had known.

Incorrect: I wish I would have known.

Correct: I wish you had told me.

Incorrect: I wish you would have told me.

Correct: We wish they had been honest.

Incorrect: We wish they would have been honest.

So this theory, in a nutshell, says you can't use "would" part with the "if" clause, rather it should be used in the result clause. So, according to this theory, your second sentence stands incorrect.