Hi kjoni,
It seems that you're working with embedded marketing form on external website.
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Required settings for prefilling feature:
-> Landing page:
1. Mark the Enable prefilled forms check box in Domain Authentication.
2. Enable prefilling feature for both fields and forms.(Remember Me is also required)
Fields level:

Form level:

You can full details about how to enable prefilling feature for landing page form:
https://docs.microsoft.com/en-us/dynamics365/marketing/form-prefill#landing-page-forms
and check this section again for how to enable the feature for embbed forms:
https://docs.microsoft.com/en-us/dynamics365/marketing/embed-forms#enable-prefilling-on-embedded-forms
Q: How will contact see prefill information for Landing page?
A: If above settings are well configured,
when he/she submitted form successfully(with Remeber Me enabled), then script in generated code will create cookie in user's browser,
by the next time contact opens marketing form again, the script will prefill fields based on saved cookie.(or refresh page multiple times)
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-> Subscription center
1. Same as lp-1
2. We won't need to set fields, form or remember me. All of these items are set to Yes by default in Subscription center.(and won't change even if we set them to No)
Q: How will contact see prefill information for Subscription center?
A: By opening subscription center link in sent marketing email.
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For your final question that what difference between those two scripts,
actually code in Form Page is almost same as code in Marketing Website if you copy them out and format them in code editor.
(combined vs separated)
Form Page

Marketing Website

So would your two scripts refers to screenshot 2?
If so,
JavaScript code is for contact's behavior tracking.
Form capture code is for form generating and data capture.
You can find answer in this link:
https://docs.microsoft.com/en-us/dynamics365/marketing/register-engagement#find-or-create-tracking-and-form-capture-codes-for-your-website
(In addition, you won't need to add form to whitelist for both landing page or subscription center if you've verified your domain)
Regards,
Clofly