These are two words that seem like they are basically interchangeable at most levels, as they generally are. The difference between them appears to be small, but what is the difference, if one exists? I feel like there are certain situations in which you would use 大事 and certain situations in which you would use 大切. For example, I would associate 大切 with sentimental things because of the 切 in it that comes from 切ない, things like memories, or a present that someone gave you, whereas I would associate 大事 with things that are of pressing importance, closer to 重要. But this does not appear to be the case, as I occasionally see 大切 used where I would normally expect 大事, like 大切なお知らせ, with more of a feeling of 緊急 or urgency. There are sources here and here which address this, but they both arrive at the same wiggly uncertain feeling that 大切 is more emotional and 大事 is more about urgency while admitting that in just about any situation the two can be used interchangeably.
And in the following for 大事:
This lends something to the idea of 大切 as a more 'polite' form of importance while 大事 is more about the concept that something is fundamentally necessary and in that regard important. However both of the entries generally seem to be nearly the same with different wording.
Is there a practical difference, and if so, what is it? Does using one over the other have any impact on the actual meaning of a sentence? When can they not be used interchangeably (outside of set phrases like お大事に)?