Which is the better deal and why for these two laptops? i7 processor with 8gb ram - $500 OR i7 processor with 16gb ram - $580?

i7 processor with 16gb ram - $580

You get double the memory for only $80. That's about the same price for buying 8 gb of RAM and opening up your laptop to install yourself.

More RAM memory is good if you run a lot of programs at the same time. IE. lots of tabs open on your web browser.

You have to look past i5 or i7. The first number in the model specifies what generation the chip is- 4th generation, 7th generation etc. Not all i7's are equal.
https://en.wikipedia.org/wiki/List_of_Intel_Core_i7_microprocessors

Without more information I can not guess at all the differences between the laptops. You have told us that they both have processors that are in the i7 family. But that is not a specific processor but a family of them, so they could vary in their speed. No clue hat they have in the way of graphic card, or anything else.

The only other thing you told us is that the second has twice the RAM memory in it. You did not tell us the speed of the memory in either. The difference of price ($80) is about right for having double the RAM in the unit.

One thing to keep in mind. Your processor is integrated into your system board. You can not change it. Your graphic card is integrated into your system board and you are stuck with it. You can easily upgrade the amount of RAM. When purchasing a laptop give preference to a better processor and graphic card. You are stuck with them. The memory you can always increase later.

$80 is a good price for an ,,8 gig ram chip.
I'd pay the extra $80