Principia is the spiritual successor to Apparatus, being originally referred to as "Apparatus 2: Principia". There are a lot of similarities between the two, but Principia is significantly more advanced than Apparatus and behaves differently in some ways. In general it is a lot more stable and has less bugs that used to plague Apparatus.
This page contains information that would be useful for Apparatus players wanting to try out Principia.
When selecting the level type in the sandbox, Principia's custom level type is the equivalent to the Apparatus level type, and the puzzle level type is equivalent to the Puzzle level type.
The puzzle level type allows you to come up with whatever win condition you choose, which can be but is not limited to the classic ball-in-basket goal of Apparatus. See Notes/Puzzles for more information.
Principia has a significantly larger selection of objects available at your disposal, including advanced signal circuiting and even Lua scripting. However, the objects that existed in Apparatus still exist in Principia in some form.
IN0 socket to make it win the level when a ball rolls into it.The level size is very configurable in Principia, as compared to Apparatus where the level is limited to a certain size. Objects can fall out of bounds in Apparatus, while with most backgrounds in Principia there are wooden walls to keep everything inside. To simulate an Apparatus-like level size you could increase the size to e.g. 300 in each direction to get a reasonably large level size.
Principia comes with a great deal of quality-of-life improvements when it comes to building levels. This includes a dedicated "construction"/"orthographic" mode which can be toggled to get a fine-grained grid control over the placement and rotation of your objects. If you are on a touchscreen then the two-finger salute to align against a hidden grid from Apparatus still exists however (Desktop players can hold down the Shift key to get a similar behaviour).
If you are on a computer then you should read through the Sandbox section of the PC Controls page, as there are very powerful keybinds to quickly do particular things in Principia like duplicating objects with the comma (,) key.
Joints and connections are indestructible by default in Principia, as compared to Apparatus where connections always have a risk of breaking. Under the advanced sandbox menu options there's the Connection Edit mode which allows you to change the strength of individual connections, or you can use the Multi-Select configuration menu which can change the connection strength in bulk on a selected group.
Cables cannot be ripped out of their sockets in Principia, and there is no way to change the behaviour to that of Apparatus. To make something more destructible you may choose to replace Signal Cables with wireless Receivers and Mini transmitters.
In Principia, you can customise RC widgets as compared to in Apparatus where they are hardcoded depending on the amount of panel sockets. You can choose to recreate the setup of the Apparatus panels, but you might want to create a more unique control scheme anyways.
Interactive objects in Principia behave differently to how they do in Apparatus. The red cylinder in Apparatus is able to apply seemingly infinite force to follow the finger, while interactive objects in Principia may be dropped if it can't follow the cursor. By default interactive objects need be within range of a Dragfield or the unlimited interactive level property enabled.
The win condition for puzzles in Principia can be all sorts of things, as you can control when and how the player wins the level. For an Apparatus Challenge-style level, you may want to use a Ball or a Metal ball combined with an Object field to detect it, wired up to a Game Manager's IN0 slot to win the level. An example can be found in this Principia recreation of Apparatus level 34.