It's the same between these two words in this case. If you want express "take" a vehicle like bus or plane, you can use both of them. And we usually use the word "搭" with "乘". Ex: 我们是"搭乘"公车过来的。 Most of time, we may omit the "乘", just say 我们是"搭"公车过来的。 Of course you can also say 我们是"坐"公车过来的。 But we use "搭" more than "坐" : "搭"飞机、"搭"船、"搭"出租车、搭地铁。 If you want to say someone "sit" on the chair, you can only use "坐" but not "乘"。 In short, “乘” is similar to “take” in English, “坐” is similar to “sit”. Anyway, if you want to express "take a vehicle", you use搭>坐>(搭)乘(more formal). I hope you understand what I’m talking about. My English is terrible!

there is no big difference between them,we use 坐 more on talking.I suggest you use "我们是坐公共汽车去还是坐出租汽车去?"

乘 for Transport 坐 for more widely usage

There are no differences here. Whether people use ‘乘（坐）’ or '坐' depends on their language habits. '乘坐‘ is more formal than ’坐‘. Hope this helps!

the two expression is useful here,there is no difference here ,however，we always use "骑单车，骑马”等