Adobe Camera Raw (ACR) is a plug-in for Photoshop (and a technology built into Lightroom) that is used to convert camera raw files into image files. The process of turning a raw file into image informaiton is called "rendering". Because the versions of ACR in Lightroom and Photoshop do not match up, when you try to send a raw file from Lightroom to Photoshop it wants to know if you plan on using Lightroom to fully render the file for use in Photoshop. Your other choice is to send the original, unrendered raw file to Photoshop so that Photoshop's version of ACR can do the rendering. Either way, your raw file remains unchanged, as always.
In these kinds of cases, generally, the best idea is the use the latest version of ACR to do the rendering. So if Lightroom is more advanced, then use the "Render using LR" option. Ultimately, the choice is up to you, there isn't really a wrong one.
Render in Lightroom:
from the link below: "render using lightroom' uses lightrooms own processing engine to render a new file which is automatically used in PS. this does not mean that an additional TIFF or PSD file is created, depending on your preferences, however all of your Lightroom adjustments will be applied correctly"
it goes on to also discuss "open anyway".
Simple anwser: Render using LR means, let LR process the raw data. Edit in Photoshop: Let whatever version of Adobe Camera Raw process the data. If the two were on parity (same version, same features and thus same processing), no difference. You wouldn’t even get that message. But ACR is older than LR so LR is telling you this. Newer functionality in LR isn’t available to ACR. In such a case, you want LR to process (render) the data. Just use the Export command and setup what you desire in terms of size, color space, file type, just as you setup your Edit in Photoshop preferences.
Adobe doesn’t force you to have ACR and LR on version parity which is nice. And if not, you are warned of this disconnect in processing the raw data. You could have the older version of ACR do the processing but there is a possibility there will be a visual difference depending on what is ‘missing’ in the functionality in ACR.