If you’re working with a mismatched ACR version (i.e. ACR 5.7 in CS4 with Lightroom 4.0), you should see the ACR Mismatch dialog, which allows you to choose how to handle the file. There’s a diagram of the options and their results at the end of this question.
Render using Lightroom uses Lightroom’s own processing engine to render the file, which is then automatically opened into Photoshop. This does mean that an additional TIFF or PSD file is created, depending on your preferences, however all of your Lightroom adjustments will be applied correctly.
Open Anyway ignores the mismatch and passes the image data and settings to Photoshop for ACR to process, which may produce something close to the correct rendering or may be completely different, depending on how great the mismatch is, and which sliders you’ve used. It doesn’t save the TIFF/PSD until you choose to save the changes.
If you choose Open Anyway but support for your camera was added after that ACR version was released, Photoshop will open but the photo won’t open, as ACR won’t know what to do with it.