They both have the meaning of"write letter to mom",but sentence 1 has another meaning ——I help mom write letter to others.

意思完全相同，这两句的主干都是“我写信”，“我给妈妈写信”中的“给妈妈”是句子的“状语”，而“我写信给妈妈”中的“给妈妈”使句子的“补语”，也就是“给妈妈”在句子中的成分不同。Mean exactly the same, this two sentences backbone is "write me", "I gave my mother wrote to" of the "to give mom" is the sentence "adverbial" and "I write to mom" of the "to give mom" to make the sentence "complement", namely "to give mom" in the sentence component is different.

Both have the same meaning and both are OK in Mandarine.

我给妈妈写信is the right way 我写信给妈妈 looks like tanslation from some software, I write to mama. that is not the right order in Mandarine, but it is okay in some dialect.

Yes, They have the same meaning. There are some additional comments I gotten from other forum as following (translated from Chinese): The core element of the sentences is "我写信", the object of writing is expressed in different ways. Example 1 is before "写信", it is adverbial; Example 2 is after "写信", it is a complement. Both are correct. The ambiguity of Example 1 is come from the diffrent meanings of "给", not from the sentence structure.