In reply to Kike Kikon:
As Rob says, I've got both.
However I've only recently been using the BD Storm 500R, and have not had the Actik Core out of the house yet, so I am not going to be as helpful here as you might've hoped. To give you a really meaningful insight into how each performs relative to the other won't be possible until each has had a lot of use, so ask me again well into 2023...
However I can say a couple of things up front:
They're a pretty similar weight and size: Storm 100g, Actik Core 88g
Out of the box the Storm feels more robust whereas the mounting of the Actik Core seems quite plasticy (ok they're both made of plastic). We may only find out which is actually tougher after a lot of use, and assuming we manage to break one of them. I've broken both Petzl and BD torches in the past, but neither in recent years.
Actik Core is IPX4 water resistant - good for splashing water, while the Storm is IP67, which means it's rated for temporary immersion. Assuming both work as advertised then each should be fine in rain, but in a proper deluge I might be happier with the Storm.
Storm 500R has an inbuilt 2400 mAh Li-ion USB-rechargeable battery. If you want a torch with the option of using AAAs then in the Black Diamond lineup it's the slightly less bright Storm 450, compatible with standard AAA alkaline batteries or BD's li-ion BD1500 rechargeable battery pack. Actik Core works with Petzl's (removable) 1250 mAh Li-ion Core battery pack, or 3x AAA batteries of your choice. The Core battery comes included in the price of the torch.
Output claims are a bit all over the shop, so it's hard to be definitive. Storm 500R is said to give you 120m distance at its 500lm max, whereas the Actik Core offers a 110m beam at 550lm. I tend to suspect the plain lumen figure is a bit like megapixels in cameras, a nice headline but not in itself definitive: surely the beam pattern and lens quality also has a part to play in how far and how usefully a torch illuminates. I will get them both out in a field to test a measured distance at some point.
Burn times on paper are similarly hard to compare. The Storm 500's figures are a lot more impressive (max power 7h vs max power 2h), which may be because of its beefier battery. Then again, I'll only have an opinion on relative burn times when i've used both in various settings (cold weather for instance)
Hope that's marginally helpful?