Gfilui

on my Desktop I have a Broadcom Wifi adapter, when I boot from Live USB 18.04LTS I go to Software & Updates, then Additional Drivers and enable it, straightforward and easy. If I install it then on the SDD (with the option of downloading restricted packages and hardware support and Wifi enabled(meaning it can download all it needs))) I end up with a system without the wifi drivers. The same Additional Drivers enabling doesn't work.

Running
cd /cdrom/pool/restricted/b/bcmwl
sudo dpkg -i bcmwl-kernel-source*

to install the driver from the Live USB fails because of dependency issues and the Ubuntu Docs are outdated, i.e. folders they mention don't exist anymore on the Live USB.
Ubuntu Docs

So it works in the Live USB but doesn't in the installation, how can I fix it?

I had a similar issue trying to get my HP Stream 11 (model: 11-r015wm) netbook connected to WiFi.

lshw -C network told me it was a Broadcom BCM43142 wireless card. Come to find out it's a WiFi + Bluetooth combo card with (anecdotally - I tried about 6 different Linux distros) not a lot of support out of the box.

I settled on using Ubuntu Budgie and similar to your experience, when running from the live CD, I could get the wireless card to work if I manually downloaded and installed the driver and some dependencies:

bcmwl-kernel-source, linux-libc-dev, libc6-dev & dkms .deb packages. It was like magic when on the last command sudo dpkg -i bcmwl-kernel-source_6.30.223.271+bdcom-0ubuntu4_amd64.deb my WiFi turned on and starting working.

I naively thought all I needed to do now was to install the OS from the live CD and install those same .deb packages. With my new and working internet connection I installed the OS on my hard drive and made sure to tick off the box about installing updates and third party software/drivers.

However, after I finished the installation and restarting, this time around when I attempted to install dkms it said I was missing dpkg-dev, make & gcc. Wait what? Perhaps the live CD contains more software or is more robust than a standard installation?

Anyways, I started to manually install each dependency as they came up by going to https://pkgs.org/ and transferring over the .deb files with a USB. By my fourth transfer I was pretty salty.

This is what I did to finally end up solving the issue:

1) Found and downloaded my driver with all of its dependencies to a USB using this method:

https://stackoverflow.com/questions/22008193/how-to-list-download-the-recursive-dependencies-of-a-debian-package

1.5) Then I attempted to install all of the .deb packages and resolve dependencies by adding a local directory to /etc/apt/sources.list.

reference:
https://www.linuxwave.info/2009/04/adding-local-directory-to-apt.html

I then ran sudo apt-get update --allow-unauthenticated

reference:
Force update from unsigned repository Ubuntu 16.04

However, my package dependencies were still not resolving correctly. I was trying to run sudo dpkg -i *.deb on the folder where I saved everything but dpkg wasn't sure about which order to process all my .deb files in. So I tried something else instead..

2) I copied all of my .deb files into /var/cache/apt/archives and followed the answer here:

https://unix.stackexchange.com/questions/366488/how-to-use-dpkg-i-to-install-package-and-dependencies-where-dependencies-are-al#366886

by running sudo apt-get install -f it installed all of my dependencies and my WiFi finally turned on.

What a mess! And I'm quite surprised about the inconsistency between booting from a live CD vs installing and booting from my hard drive.