The Difference Between The Standard & Core
There are only two main differences between them:
1. Brightness and maximum beam length.
The Actik Core has 350 Lumens (vs 300 on the standard) which gives you an extra 5 meters (about 15 feet) beam when you are using it on maximum.
2. Rechargeable Core Battery.
Both of these lights can use the Petzl Core rechargeable battery, but the Actik Core comes with one in the box.
Why are these batteries so great? They fit in the standard compartment, so it's easy to swap in/out with other batteries you might need (spare alkalines are always worth having on hand).
Also, the battery itself has a built-in USB charging port. So, you just have to open the battery compartment and plug it in while it's in the headlamp. Very little fuss.
Which Headlamp Should You Buy?
The only other reason to buy the Actik Core is if you also want a little extra max beam reach and brightness. It's not much at 50 Lumens, but every bit counts in some situations (mountain biking, trail running, night skiing etc).