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It is both climbing, but there r huge differences between technique, equipment, difficulty, etc. You may be very good rock climber, but that wo not help u much on the ice. First of all, the danger is greater (due to fragility of ice). You have to ''nail'' the ''safety wedges'' by yourself, & leave at least one in the ice (unless if it is very popular & often climbed route). It takes extra skills to master the crampons & ice-axes well. It also take years to learn to recognize the proper ice (weather conditions may spoil it very quick). You also have to pull some extra 5-10kg with you, compared to summer rock climbing. It is also more expensive. But I'd say the reward comes in ''crystal''. :)
So, not so many similarities. Maybe the only one is ur physical & mental condition. Same muscles work out, & similar emotions run through ur head.

You CAN use all the same equipment, except for the boots & crampons, ice axes & protection. If u have the money, u can buy more specialized gear, for example double ropes r the best for ice climbing! Also, separate ice & rock ropes r a good idea because rock climbing takes the waterproofing off the rope, which makes it pretty useless for ice climbing. For protection u use ice screws instead of cams or nuts, but u can use the same slings & quickdraws to clip into. However, if u have the money it is nice to get a set of draws with big, wire gate biners for ice climbing, because they're easier clip with frozen hands & gloves, & also if they get iced up it is easier to open them.
They also make specialized harnesses for ice climbing, but u can also get them suitable for rock & ice, it just needs adjustable leg loops to compensate for different amounts of clothing from winter to summer. For helmets, it is the same thing, u can use just one helmet, but some prefer a more protective helmet for ice climbing, & a more minimal helmet for rock climbing.

Almost nothing is the same. You could use ur belay device, rope, & helmet. thats about all l can think of. In mixed climbing ice/rock, u can use some regular rock pro. Ice climbing u will need a dry rope or dry twin rope. ice ax, crampon compatible boots. climbing crampons( they r different then walking crampons). ur protection is usually ice screws & the occasional snow anchor. & then the clothes they r usually warmer & more waterproof than rock climbing clothes. so l would say almost nothing is the same

Carabiners, runners, harness, belay device, and possibly even the rope if you have both in mind when you choose your gear.

Good Luck

Well l only went ice climbing for the first time this past easter so l am not really one to go into much detail about gear. The previous seem pretty good. l found that the guy l was climbing with, who is a less experienced rock climber, but much stronger & fitter than me was struggleing more than l was. So l guess the techniques u pick up rock climbing really help u more than u might imagine when Ice Climbing. l liked to just find little pockets in the ice with the pick & not even hammer it in. Saves so much energy, but when you've got ur ballance & confidence down it works the charm.