Big star weapons are usually more powerful, carrying +54 bonus damage to the low base damage once you use them enough to max their compatibility(which takes longer to do than normal weapons). The icon shows what class of weapon they are and any character that uses that class (like spear) can use that weapon also. Though I haven't bothered with them much to really figure out if and how they are shared.
4-star normal weapons are generally less powerful in overall stats, though I have read that it is possible to get ridiculously lucky farming them in gauntlet mode using certain skills to earn versions that max at +99 with crazy skills, but that is out of reach very endgame stuff I have not seen yet for crazy min/max players. Generally big star weapons are superior to 4-star ones in raw damage numbers normally, but take longer to build up and start off weak.
Mystic weapons are 5-star weapons that are better than both of the above and can ONLY be earned by special character challenges in unique stages. Gotta google a FAQ or guide since this game does not hint at how to find them. If you use recommended teams in missions though you may stumble on completing some challenges naturally, if you are playing hard or above difficulty. Cannot be earned on normal.
Finally there are the -X versions of weapons, that are the best in terms of raw damage, and can only be crafted by a blacksmith using special materials found in gauntlet mode. I haven't bothered making any yet, I much prefer earning mystic weapons anyway as I enjoy the exclusive challenges for each character. But you will see these available to craft as you start earning the materials in gauntlet mode.
TLDR: 4-star< Big-star< 5-star< 'X' wpns.
