Hi julianE,
So, I installed easyEDA and gave it a test drive. Here's my (limited, new to EDA in general) comparison of the two programs. For the most part easyEDA and CircuitMaker are very much the same. Both are online systems. This provides team collaboration, version control and access to community component part libraries. This also means you can't work offline with them. But, that's the price you pay for the tools being rubust, and free.
The only real differences that I noticed between the two programs were in the available component 3D footprints from each community library. CircuitMaker seemed to have more 3D objects in their libraries than easyEDA. Most of the components in the easyEDA libraries just had 2D footprints (ie. those that define component pin placements and an outline of the part on the silkscreen layer). So right off the bat when you visualize your easyEDA circuit board in 3D, you only see the board. I created a simple circuit (3 pin connector, a TO-92 transistor, a couple of 1/4 watt resistors and a .5 mm LED. When I did a 3D view, only the transistor had a 3D footprint on the board. If I wanted to see the resistors, connectors and LED, I would have to find or build my own 3D files for them. Now, I didn't do a deep dive into all the community library components for easyEDA, so there may be 3D models out there which I didn't find with a quick scan. Additionally, easyEDA uses an .OBJ or .WRL 3D object file format. These are general purpose, kind of universal 3D object file formats, where CircuitMaker uses the .STEP (Standard for the Exchange of Product Data) file format, which is kind of an emerging industry standard for component part 3D objects.
Except for some syntax and naming convention differences they both otherwise operate exactly the same.
Both of these programs require strict adherence to a development process that starts with first laying out a schematic diagram of the circuit that you want on your PCB board. You need to have EVERYTHING that you want on your circuit board represented on your schematic diagram, board edge pads, header pins and connectors, resistors, capacitors, diodes, LED, transistors, regulators, ICs, etc. etc.). These objects must come from the community libraries, or your own creations (that get submitted to a community library). You need to pay attention to each component footprint to make sure it fits your needs (surface mount, through-hole, small, medium, large, etc.) There will be several versions of a particular component in the libraries to pick from. Then you need to connect everything together in the schematic with a "wire" or "connection" tool. This is a drafting-like process, but you need to make sure that the "wire" gets connected (snapped) to component leads so the software knows the paths between components. The software will indicate an open connection as an error. Once you are pretty sure you have your circuit properly laid out on the schematic you can then transfer that info over to a PCB layout window. There all the part footprints get piled into a lump with rubber-band like connections that represent your schematic. Over there in that window you define your board dimensions and start dragging your component footprints around to create your PCB layout. You'll see those rubber bands stretch around as you place your devices. The rubber bands help you visualize the circuits. Then you have to create the copper paths between component pins. The rubber-bands will snap to your copper paths if you lay it out right. The software has rules that warn you when you have a crossed connection or you are too close to another path or pin. For a double-sided board you will be on the "top layer" and "bottom layer" when creating your copper paths. The minimum layers needed for Gerber files are Top Overlay (the top silkscreen layer), Top Solder Mask, Top Layer, Bottom Layer and Bottom Solder Mask. The Solder Mask layers are automatically generated, as is the Top Overlay for the most part. You will also need a drill map, which is often created separately. The drill map is not a layer file, but the map file must be included in the Gerber files to get your holes drilled.
Whew! And this is actually just a short overview of what I discovered in creating a circuit board to be manufactured. It's no simple task to start with a hand drawn schematic and end up with a circuit board design laid out in Gerber files.
Every time I get my hands on a new piece of development software (from Photoshop to Vegas Studio to Visual Studio to Blender, now EDA) I see how whole professional careers can be had using any one of these tools.