"He acted strange(ly?)"

It would make sense if both of these sentences were grammatically correct; but is anything different between them meaning-wise?

He acted very strange when I told him about the missing amulet.

He acted strangely about the whole deal.

What difference is there between using an adverb here and using an adjective?

Acted is a verb which can take an adjective in what looks like a modifying position. Some verbs are like that; they are called copular (or copulative or copula) verbs. Let's consider the verb look, where using an adjective and an adverb produce different meanings.

For some meanings, you have to use nice:

John looked very nice in his new clothes.
*John looked very nicely in his new clothes.

For others, you have to use nicely.

John looked at me very nicely.
*John looked at me very nice.

In the first example, very nice is in some sense modifying John, while in the second, very nicely is modifying looked.

For the verb act, you can use either an adjective or an adverb:

John acted very strange.
John acted very strangely.

and the meaning isn't any different between these two sentences (or at least not much). I would say that in the first sentence, John is acting as if he was very strange, and in the second sentence, John's actions are very strange. In this case, the meanings end up being the same. But consider:

John acted very quiet.
John acted very quietly.

Now, these mean different things. In the first, John is simply being quiet. In the second, John is doing something and trying hard not to call attention to it.

It is not uncommon for people to use the adjectival form of a word where you might expect an adverb. Indeed, this is one of the biggest sources of confusion for native and non-native speakers alike.

Certainly you could say, without triggering the Pedant-O-Meter:

He acted hostile toward me.

whereas

He acted hostilely toward me.

would trigger comments from just about anybody, although the sentence does seem to require an adverbial form.

Now consider the following:

I felt bad for you.
I felt badly for you.

Many people prefer the latter, thinking it's grammatically correct, but the fact is it literally implies some kind of defect on the part of the speaker's ability to feel. Turn it around: Do you ever feel "goodly" for someone?

To sum up: Informally, you can use either, and most people would say a person acted strange (in everyday speech). In more formal situations, it would be prudent to use strangely instead.

This is an idiom that seems common in US English. Speakers of that dialect commonly substitute adjectives for adverbs. There isn't any extra meaning attached to this, it is just a dialectic peculiarity.

In fact, in the US use of an adverb in some situations sounds a little formal. For example:

This is real good coffee

This is really good coffee

You will hear the first commonly in the US, and the second sounds formal. In the UK the first sounds "American" and the second is perfunctory.

I don't know about other dialects, US and UK are the ones I am familiar with.