I have all was been a fan of ice theme champions and have been think on picking up a mage. The thing is League have two ice theme mages. 
and 
. I have play both of them before and I do like both of them but I wish to try and main one of them and only one.
So I wish to ask others on who you think is the better ice theme mage. The Cryophoenix or the witch?
You want to offer burst or utility?
Lissandra has tons of CC and can tell an assassin to sit down and take a time out while your team finishes grown up business.
Anivia clears waves all day and pops squishies like wuuuuuuuuuuuuut.
Both are hard to tower dive. Anivia almost can't be dived when egg is up, and lissandra is almost impossible after level 6.
Both have different strengths and weaknesses so I guess it depends on how you want to play.
Liss generally has less range than other mages but can do really well in teamfights. She does pretty well top too against melee champs, since she can get pretty tanky and has ways to keep them off of her while she damages them. Her ult is also really good at protecting you from stuff like zed, vlad, fizz, karthus ults, etc. Her passive is good at saving mana since she gets that free spell (forget the timer on it) but I feel like she depends on her team's comp more later in the game to make an impact unless she is tanky/can get close enough.
Anivia's passive is really helpful but only helpful if you're in a spot you can regenerate in. If you die in a teamfight and turn into your egg, its gonna get focused. She has a good amount of range though and can do crazy damage with her e. Her ult is good in teamfights for the stun but if she doesn't have her q up and she's being attacked there isn't much she can do. She could ult but depending on who is following it might not matter. Not sure how she'll be with the new item changes because I've heard they are getting rid of a lot of mana regen which is pretty useful on her since her ult drains mana. Who knows though, that isn't for a while.
Why not both? learn em and swap out for comp reasons.
More mobility and CC. Is kinda boring and linear IMO.
I like Lissandra so much better, and thematically she's one of the strongest League champions.
She's an anti-assassin mage, so she does not have mobility and most of her abilities are laced with CC. You won't be picking up a crazy amount of kills as Lissandra IMO, but you also won't be dying nearly as much. She's low risk, and this is ideal in ranked.
Lissandra has many abilities to help make up for her lack of mobility and short range, and I don't think a team can ever have enough lockdown, so she works in a lot of team compositions.
The one downside to Lissandra is that she's incredibly susceptible to poke. It's funny you list Anivia, because that's a lane matchup that does not favor Lissandra. Even so the likelihood of them killing you is not extraordinarily high.
Lissandra can also be played in top lane, but I don't think it works as well in the current meta. Top lane is pretty out of control at the moment, and Lissandra is again struggling to find her place in the meta, but I still believe she's a solid choice.
