Learn English – What’s the difference between “I do know” and “I know”

Level of knowledge.

I know the C language. I can write programs in it proficiently. I know the caveats and problems.

I know about Python. Basics of syntax, general structure. Given some time and references I might write some simple programs in it, but saying "I know Python" would be too much.

I know of Haskell. There is such a language and it's a functional language. That's about all I know of it.

Virtually means "almost (in respect to degree or strength)" or "almost completely".

In your first pair, the difference is very small:

This mission is almost impossible to achieve.

This mission is virtually impossible to achieve.

Here, I think virtually is a stronger choice, so the second sentence simply means the mission is more difficult than the first. Both sentences mean basically the same thing.

Your second pair has a much bigger difference:

He was almost killed in that accident.

He was virtually killed in that accident.

Here, the first sentence can mean either:

• He was almost hurt in an accident, but he wasn't hurt at all. (For example, his car almost fell off a cliff, but it didn't fall. He got out safely.)

• He was hurt very badly in an accident. He almost died from his injuries.

On the other hand, the second sentence (with virtually) can mean only the second sense. He was hurt badly and was close to dying.

Perhaps a good rule is to imagine that the adjective modified by virtually really is true, but then take a tiny step back:

He was killed in that accident. He's dead.

He was virtually killed in that accident. He's been in the hospital for a month.