Fill in the blanks:
The phone is ringing, ...
1.I'll pick it up
2.I'm going to pick it up
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Often the two expressions get used interchangeably, a lot depends on how it's said and surrounding context.
However, without anything else, in your example with a ringing telephone
I will pick it up.
I'll get it [more idiomatic]
would mean you will answer the phone now and has more immediacy and determination than
I am going to pick it up.
which might mean you are about to start to go pick up the phone.