using the verb "blame"

I am not to be blamed

Many accidents were blamed on the snow

be to blame

If a ship sinks, the captain is to blame

The increase in crime is a sad reflection on our society
The increase in crime is, sadly, to be blamed on our society

I am wondering about how is the function of applying the verb blame, that is, I cannot differentiate when to use it as an intransitive, or passive verb.

Furthermore, I am wonder if the bold parts might mean the same thing or not.

Answer

The word blame is used to reflect responsibility for something.

To answer your question about the bolded sentences: "The increase in crime is a sad reflection on our society," holds a nuance that suggests the crime is increasing because society is lacking in morals. There is no implied 'call to action' in that sentence.

In the second case: The increase in crime is, sadly, to be blamed on our society, the sentence uses the word blame which implies a sense of responsibility. Society is responsible for the crime, and it then follows that society is responsible to do something about the crime. This sentence creates a feeling of requiring action.

The word blame is always used transitively like the word "hit". You have to blame someone/something. Blame is used passively in writing, such as the examples that you mentioned. In modern American English, it is rarely used passively ("be to blame") in speech. It would be used with be ~ing instead:

"What happened?"

"They're blaming the captain because the ship sank."

"Oh! I thought they were blaming the ship company instead."

It is also possible to use blame actively, but it's quite rare. In fact, I can only think of it being used in a very direct way. ie. "I blame you/him/her." To which it would (typically) be unnatural to reply, "Why do you blame me/him/her?" You would have to respond, "Why are you blaming me/him/her?"