Both are relatively slow, specially offensive Pokemon.
Mr Rime has higher speed and special attack, but lacks the addional bulk that appletun provides.
It would be easier to say that you should choose based of whether or not you want bulk or speed, but it's not that simple.
Appletun has good defensive moves in Leech seed and recover, but lacks toxic. Any competitive player knows the significance of toxic on a defensive/stall Pokemon.
Mr Rime has excellent coverage.
Thanks to it's offensive typing it can deal super effective damage to 5 types.
With dazzling gleam, energy ball, focus blast, thunderbolt and shadow ball included, this total increases massively.
Mr Rime has great special attack especially when supported with life orb, but since it's not bulky and has rather average speed, it can't make use of it.
Appletun might seem like the better option, but it's coverage makes the decision hard. With coverage mostly catering to it's physical attack, which is lower than it's special attack, it's hard to get appletun good coverage.
Since PU is not the most powerful tier, both Pokemon, who are obviously not that good, might still have use.
I would pick Mr Rime, but then another problem rises.
Absol, Aggron and Archeops, can all deal offensive damage to Mr Rime, while Archeops and Absol outspeed and have priority.
These are just a few of the challenges that Mr Rime is up against.
Appletun has it's weaknesses, but they are not as extreme thanks to its better defensive typing and defensive stats.
Overall, I'd choose Mr Rime and pair it up with a Pokemon that I can switch out to.
The offensive ability of Mr Rime is something much better than the defensive or offensive ability of Appletun.
