The way I finally got through my head the difference was to think of the cutter as a pinch roller that's trying to move your work along. With conventional milling, that pinch roller is working against you. With climb milling, the pinch roller wants to pull the work deeper into the cut. Think about it that way as you are feeding--does the pinch roller pull in the direction of feed or push against?
The safety is a function of your backlash, and the ability of the "pinch roller" to overcome your machine's friction. If there is no backlash (for all intents and purposes) as with a ballscrew on a CNC, it doesn't matter. If you can successfully increase the friction enough (by running with very tight gibs, for example), you MAY create a situation where the "pinch roller" (your trusty milling cutter) can't generate enough force to overcome the friction, in which case you can climb mill there too.
If you are unfortunate enough to climb mill with significant backlash and not enough friction to offset the cutter force, what will happen is your work can get "sucked" into the cutter a distance equal to the amount of backlash. Depending on the machine, cutter, workpiece, how well it is clamped, etc., etc., this can result in problems ranging from a requirement for new underwear to serious bodily injury. Have you ever seen a surface planer in a woodworking shop throw a plank? Or perhaps a table saw with a nasty kickback? Well, a bad incident here can have that drama plus shrapnel from the cutter. No fun.
Something else to consider is reducing the force the "pinch roller" can create. Use a smaller cutter and MUCH less depth of cut. Hopefully you are only contemplating a risky climb cut because you're in search of a nicer finish anyway. So if you are using a very light depth of cut commensurate with a finer finish, your pinch roller won't pull as hard.
Or, consider a heavier mill with a heavier table. There is a lot more to get moving there with a light cut. Add tightened gibs and you're getting there. Is it worth it when the handwheel is so hard to turn that it's hard to get a good finish for that reason? You gotta wonder.
OTOH, having the ballscrews and CNC on my machine, gosh you can get a nice finish.
Cheers,
BW