Paper vs online portfolio

Hey everyone!
I have been lurking and reading a lot of threads for a while now and i thought i add a question about something i have been wondering about.

should your paper portfolio that you bring to interviews be the same as what you were to send employers through an email with your cover letter. I know that when emailing you want to impress who ever is looking at your work, so they will invite to interview. but if you get invited for an interview and bring pretty much that same thing you emailed (just printed) is that bad?

I have a portfolio website that i include a link to in my cover letter that has most of the same projects as paper portfolio.
My paper version goes a bit more in-depth and is laid out nicely on a page. but employers dont see this version unless they like the downscaled version on my website. but if you are trying to impress them through an email wouldn’t they want to see the good stuff?

Im sorry this doesn’t make sense, i am pretty confused about the situation.
basically what i am asking is should you send your full portfolio to a potential employer?

The portfolio you bring should be more in depth. Use it to stoke the conversation longer. Make it as much of a two way conversation as possible.

The portfolio you show on line or send in should be as visually powerful as possible.

They each have a different purpose so they require a different design approach.