Speaking in Illustrator terms....
Open (Vector file): If you open a vector-based file which Illustrator supports you have access to all the traditional editing features for vector artwork - paths, anchors, etc. Opening a vector file with Illustrator is pretty much the same as if you were to create a new file and draw. Any restrictions when using Open would depend on the specific file you are opening. For example, if you have an SVG file you can open that with Illustrator. However, some SVG structure may not be well supported in Illustrator so it's feasible that the artwork may not be complete using the Open command. Illustrator will traditionally alert you if there is a problem when opening a file.
Open (Raster file): You can "open" a raster based file with illustrator as well - jpg, png, psd, etc. When you "open" a raster based file, which is not a .psd file, you essentially get an embedded raster image. (See below for more on embedding).
When you open a layered .psd with Illustrator you are presented with Photoshop Import Options, to use the layers in the .psd. This will allow you to retain some layer edit-ability within Illustrator, but only for .psd files.
Place: creates a link to the file being placed. See below for link explanation.
Link: When you link to a file, vector or raster, the output data of that file is placed into Illustrator as a reference to the original file. That means if you link to "FileA" in Illustrator, then you go change "FileA" later, the link will ask you if you want to update it to reflect the file changes. Linking to a file creates a connection between the document you are working on and the file being linked to. This allows you to include external files within your Illustrator document without the need to copy everything into the document you are working on. Linked files are not directly editable within Illustrator. You can't change paths/anchors in a linked vector file. And you can never change the pixel data of a raster file in Illustrator.
Embedding (vector file): When you place a vector file into a document is is placed as a linked file (see above). You can then choose to embed that linked file via the Links Panel. With a vector based file, embedding breaks the link to the external file and places all that file data into the current document allowing traditional vector editing of the data. There's no more connection to the external file. This is pretty much the same as if you were to copy/paste a vector file into your working document.
Embedding (raster file): Using Place with a raster file places a linked file in the document (see above for link explanation). You can then choose to embed the raster file via the Links Panel. Embedding a raster file merely means there's no more connection to the external raster file. The raster data is copied to the current document. In terms of editing, there are a few instances where Illustrator requires a raster image be embedded to perform certain actions on the image (such as envelope distortions). However, other than those, there's no decreeable difference between a linked and embedded raster file, unlike linked and embedded vector files.