If someone who doesn't really know German says
Ich bin [name]
Ich heiße ...
to introduce him/herself, is this an error in German and how odd does it sound?
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The difference is in the level of formality vs casualness here.
Ich heiße Fritz Müller
would be a formal, almost stiff way of introducing yourself. Someone presenting himself to a conference auditory he is new to could use this.
Ich bin der Fritz
is a very casual (but friendly) way of introducing yourself. Fritz thus suggest to use "Du" for addressing him.
Ich bin Fritz Müller
is somewhat in-between: both formal and to some extent casual.
"Ich bin X," in this context is not wrong. It's just not formal German.
It's like the difference in English between saying "Hi," and "Hello." The latter is more common, but using the former isn't an error, or even "odd." It is just "less common" and less formal.
So the answer to your question is, "Nothing much."
When I was in school, at the age of 14, we read the book “Krabat” in the German class. One sentence I remember strongly is:
Ich bin Krabat, ein Mühlknappe aus dem Koselbruch.
We discussed the difference between “ich heiße” and “ich bin” for quite a long time in class, maybe even a whole lesson. Since then I prefer “ich bin” over “ich heiße”.
The latter sounds technical, like answering a question for specific information, while the former puts emphasis on the person as a complex being. “I am”, I identify with all of my thoughts, my traits, myself.
Most probably, people who had other teachers will have made other experiences.
There's an ever-present danger of a touch, or more than a touch, of arrogance, pride, or self-obsession being either implied or inferred by "Ich bin".
There's more control over this (for better or worse) when it's spoken rather than written. If written, the nuances in its meaning will vary according to context and other evidence.