Is the GNOME Tweaks package in Software the same as installing gnome-tweak-tool?

Yes, the gnome-tweaks-tool and gnome-tweaks packages correspond to the same application.

The package was renamed to gnome-tweaks with GNOME 3.26 (Ubuntu 17.10), but it continues to provide gnome-tweak-tool as a transitional package, which can safely be removed (is used for those who upgrade), as you can see here: https://launchpad.net/ubuntu/+source/gnome-tweaks/3.28.1-1

GNOME Tweaks is the new name for GNOME Tweak Tool as stated in the accepted answer. The release notes of GNOME 3.26 (Ubuntu 17.10) states:

• Tweak Tool has been renamed to Tweaks and has gained three new settings: a switch to move window buttons to the left or right, a Disable While Typing option for touchpads and an option to show the battery percentage in the top bar. There has also been a good amount of clean up and refinement.

For a full list of tweak tools in 18.04 (I'm using Unity Tweak Tool) run this command:

$ apt list | grep tweak
gajim-rostertweaks/bionic,bionic 1.0.0-3 all
gnome-tweak-tool/bionic,bionic 3.28.1-1 all
gnome-tweaks/bionic,bionic 3.28.1-1 all
mate-tweak/bionic,bionic 18.04.16-1 all
mousetweaks/bionic,bionic,now 3.12.0-4 amd64 [installed]
tweak/bionic 3.02-2 amd64
unity-tweak-tool/bionic,bionic,now 0.0.7ubuntu4 all [installed]