They can be "synonymous enough" in most sentences, but they don't literally mean exactly the same thing. For example, these sentences have similar enough meaning to be used interchangeably most of the time: - "I ask myself why am I doing this." - "I wonder why am I doing this." But they are imperfect synonyms, because wondering is more passive, and is a central thought process, but asking is more assertive, and is a beginning to a process of checking. For example, these sentences have slightly different meaning: - "I ask myself why am I doing this, so I wonder what the answer is." - "I wonder why am I doing this, so I ask myself what the answer is." While they have very similar meaning, the first sentence implies I begin assertively enquiring, then I become passively unsure. The second one implies I begin passively unsure, then I become assertively enquiring. Very subtle stuff, I wouldn't worry about it to be honest. You could treat they as practically synonyms, with slightly different emphasis between them.

In my opinion, "I ask myself" is not very common in spoken speech. But both have similar usages. However, the verb "wonder" is sometimes used without a quotation, while "I ask myself" is most often used with a quotation. Some examples: 1. "I wonder what is it?" I wondered. = "I wonder what is it?" I asked myself. 2. "Do you really mean that?" I wondered. = "Do you really mean that?" I asked myself. 3. I wondered about the meaning of life. For number 3, "I asked myself about the meaning of life" sounds unnatural.

Wondering involves use of the imagination. Asking is more matter of fact. However, "asking" can be softened up to make it more like "wondering" as in: "now and then I ask myself if my son will ever marry". Because of "now and then" it is the same as wondering.

To me, " I ask myself " sounds good in poetry, drama, maybe sarcasm, but probably not so good in an ordinary chat with your friends, or yourself!

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