You can download and run more than one version of MuseScore in the same computer so you will always have a choice of which version to use to open a particular score. I have a large number of scores which were created with 1.3; some of those I've 'converted' to 2.0 and others not (because they are completed and the print masters have been 'locked' in pdf form).

Converting an older score to a run in a newer version generally just requires opening it with the newer version, and then saving it. It's wise to save it under a NEW NAME, however; you can simply add 'MS2' or some similar indication to the end of the file name. That way, if something goes wrong with the process, you haven't destroyed the original file and can try again. Even safer (if you're computer-paranoid, like me) is to open it with the original version, save a copy under a new name using that same original version, and then open the copy with the new version.

On occasion, when you open an older score with a newer version you will get a warning popup stating that the file is corrupted. Most of the time it is sufficient to hit 'ignore' on that popup and the thing will open anyway, although there may be some gibberish to find and correct manually.

The danger of using a different version to work on an almost completed score is primarily editorial: If the score has been proofread and formatted for page make-up, opening it with a different version of the program than the one which created it could introduce minor changes in layout or formatting that would require you to proofread it all over again.