According to the distance measurements by the satellite Hipparcos, the (somewhat uncertain) distance to Betelgeuse is more like 500 than 600 light-years. So if Betelgeuse exploded 600 years ago, the famous outline of constellation Orion would just be missing its alpha star!
By the way, you asked what would happen if Betelgeuse exploded 600
light-years ago. Please note that light-year is a unit of distance, in the same way that kilometers and miles are units of distance.
Normally, when we discuss when a star goes supernova, we don't talk about when the star exploded "from its own point of view", but rather when
we saw (or might have seen) its explosion. For example, in 1987 a star exploded in the Large Magellanic Cloud, which is a satellite galaxy of the Milky Way. The distance from (our location in) the Milky Way to the Large Magellanic Cloud has just been refined to about 163,000 light-years. (Read about it
here.) If we were to try to decide when the star exploded in the Large Magellanic Cloud "from its own point of view", we might try to say that it exploded 163,000 years ago. But for us, it is really so much more practical to say that the star exploded in 1987.
You are really asking what it will be like, for us, when we can see Betelgeuse explode as a supernova. All I can say is that it will be a fantastic light show, but it will almost certainly be completely "harmless". For us, that is.
I'm almost sure I have read that the safety distance to a type II supernova is about 30 light-years. A type II supernova is a star that explodes because it is so massive that it eventually builds up an iron core. An iron core is doomed to contract violently and set off a chain reaction that destroys the star. This is what will happen to Betelgeuse.
But Betelgeuse is too far away to harm the Earth or us when it explodes.
Ann