Upgrading from Mathematica to Wolfram

Starting in Version 14.1, the Wolfram application was introduced as the new way for users to access Mathematica, Wolfram|Alpha Notebook Edition, Wolfram|One and Finance Platform. With this update came a number of changes that are not backward compatible.
Accessing Mathematica from Wolfram
To access Mathematica 14.1+, install Wolfram as you have installed Mathematica in previous versions. Once it is installed, the product(s) that you own can be activated in it. Follow the activation process if your Mathematica license is not automatically imported on startup to access Mathematica through the Wolfram application.
If you have multiple products to activate or a new license to add to your product, navigate to the Product Settings tab in the Preferences menu.
Product Settings/Switching between Products
The Product Settings screen is a new tab in the Preferences menu. It allows switching between products and activating additional ones. Each product/license that is locally activated will show up as a separate entry on this screen. See the tech note Using the Wolfram Product Switcher for more information on using the Product Settings screen.
Note that all products/licenses for a specific version will now share the same settings, since they are stored in a common location.
Migrating Paclets and Packages
Since it has a different $UserBaseDirectory (and associated paths), Wolfram will not automatically detect paclets or packages that had previously been installed for Mathematica (or other Wolfram products). There are a few different ways that Mathematica paclets/packages can be brought over to Wolfram.

Freshly Install the Paclet or Package

The cleanest/safest way to get a paclet or package that you had previously installed in Mathematica to work with Wolfram is to just reinstall it for Wolfram. Most paclets will just redownload as needed. Some, like ones users downloaded from the Wolfram Language Paclet Repository, will likely have to be reinstalled with PacletInstall.
After installing and activating Wolfram, you can install a paclet with one line of code if you know its name on the server:

Copy All Existing Paclets to the New Location

Hand-copying the entire contents of Mathematica's $UserBasePacletsDirectory/Repository to the equivalent Wolfram folder is the simplest way to get all of your old paclets into Wolfram. This will also bring over any old versions of paclets on your system, but they will be ignored by Wolfram and will just cost some disk space. You will never need to redownload any of those paclets, so this will be the easiest solution.
Paclets are installed, both in previous versions of Mathematica and in the current version of Wolfram, in the Repository subdirectory of $UserBasePacletsDirectory. This will be in different locations for Mathematica than for Wolfram. Evaluating:
will get you to the correct paclet location.
C:\User\ username \AppData\Roaming\Mathematica\Paclets
Windows
~/Library/Mathematica/Paclets
macOS
~/.Mathematica/Paclets
Linux
Typical default values of $UserBasePacletsDirectory for Mathematica (old).
C:\User\ username \AppData\Roaming\Wolfram\Paclets
Windows
~/Library/Wolfram/Paclets
macOS
~/.Wolfram/Paclets
Linux
Typical default values of $UserBasePacletsDirectory for Wolfram (new).

Copy Specific Existing Paclet(s) to the New Location

Hand-copying only selected paclets from $UserBasePacletsDirectory/Repository will get the specific paclets you choose copied to Wolfram. You could choose to copy over the largest ones, to save download time in the future, or only the Wolfram Language Paclet Repository ones (these have __ (double underscore) in the directory names and are likely to have been hand-installed).
By choosing the individual paclets this way, you can avoid bringing over paclets you no longer want and avoid having old versions of paclets taking up space in the new location. See the previous section for the old (Mathematica) and new (Wolfram) paclet locations.

Copy Existing Packages to the New Location

Packages generally live in $UserBaseDirectory/Applications. Like with the two preceding paclet sections, packages can be copied over individually or all together. Bring over any/all apps from the Mathematica location to the Wolfram one for the packages that you want to continue using in Wolfram.
Migrating Stylesheets
Similar to paclets, the new $UserBaseDirectory (and associated paths) makes it so that Wolfram will not automatically detect stylesheets that had previously been installed for Mathematica (or other Wolfram products). There are a few different ways that Mathematica stylesheets can be brought over to Wolfram.

Freshly Install the Stylesheet

The cleanest/safest way to get a stylesheet that you had previously installed in Mathematica to work with Wolfram is to just reinstall it for Wolfram. After installing and activating Wolfram, Select Format Stylesheet Other... to install stylesheets in Wolfram.

Copy Existing Stylesheets to the New Location

Alternatively, you can copy previously installed stylesheets from your current Mathematica installation location to the Wolfram installation location. This can be done simply by finding the proper locations and copying over whichever stylesheets you want in the new location.
Custom stylesheets for both Wolfram and Mathematica will usually be in the subdirectory of SystemFiles/FrontEnd/StyleSheets subdirectory of $UserBaseDirectory. By looking in both locations, you should be able to find and copy the desired stylesheet(s).
The new and old locations can be found in Wolfram using the following commands.
Wolfram (new) default locations:
Mathematica (previous) default locations:
It is uncommon for stylesheets to be in $BaseDirectory/SystemFiles/FrontEnd/StyleSheets, but system administrators or others may place stylesheets here as well.
Copying Preferences
It is recommended that if you want your Mathematica preferences to be used for Wolfram, you recreate them in Wolfram. This way you don't have to worry about stale preferences affecting the current version of the product.
That said, if you still want to copy your old preferences from Mathematica to Wolfram, you can grab the relevant init.m files from your kernel and/or front end locations. The locations mentioned below are for that product (Mathematica or Wolfram)to access the preferences location for the other one, update the file paths accordingly.

Kernel

The initialization files used by the kernel contain Wolfram Language commands that are evaluated at startup, such as function definitions, packages to load and kernel option settings. These commands are also available to the Wolfram System front end.

$BaseDirectory /Kernel/ init.m

This file is used to initialize the kernel and therefore should store any kernel commands that are appropriate for all users running the kernel.

$UserBaseDirectory /Kernel/ init.m

This file stores user-specific commands used to initialize the kernel.
Kernel initialization files are not changed automatically. If they have not been manually edited, they will contain no commands.

Front End

Front end initialization files store preference settings, such as the values of global options in the Option Inspector. These files are automatically updated by the Wolfram System when you change a setting in the front end. To make changes to this file, we recommend modifying settings from within Wolfram rather than editing the file directly.

$BaseDirectory /FrontEnd/ init.m

This file stores systemwide defaults used by all Wolfram System front ends. The settings in this file affect all users. "Systemwide Defaults" explains how to implement changes to this global file.

$UserBaseDirectory /FrontEnd/ init.m

This file defines settings within the front end that are specific to each user.
Front end initialization files are modified automatically as user preferences are changed while using the front end.
New Locations and Names
The executable name has changed from Mathematica to WolframNB.
Wolfram has different default values for the locations of built-in variables such as $InstallationDirectory, $BaseDirectory and $UserBaseDirectory.
C:\ProgramFiles\WolframResearch\Wolfram\ 14.1
Windows
/Applications/Wolfram.app/Contents
macOS
/usr/local/Wolfram/Wolfram/ 14.1
Linux
Default locations for the Wolfram installation directory.
C:\ProgramData\Wolfram
Windows
/Library/Wolfram
macOS
/usr/share/Wolfram
Linux
Typical values of $BaseDirectory.
C:\User\ username \AppData\Roaming\Wolfram
Windows
~/Library/Wolfram
macOS
~/.Wolfram
Linux
Typical values of $UserBaseDirectory.
Environment variables used to customize locations used by the Wolfram Language have also been changed.
MATHINIT is now WOLFRAMNB_INIT
MATHEMATICA_BASE is now WOLFRAM_BASE
MATHEMATICA_USERBASE is now WOLFRAM_USERBASE