What is the exact meaning of every sentence?
I look like my grandmother.
I take after my grandmother.
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If you look like someone in your family from whom you have descended, you can be said to take after them.
However, for example, if you have some particular talent, your parent or grandparent has, that can also be take after.
My grandfather had a talent for music and I take after him. [that is, I too have a talent for music].