When saying “I miss you” which of these is more frequently used? Do they have different nuances?
会いたい in the literal sense means 'I want to meet', it is something you would say to close friends or a partner. The meaning is closer to 'I miss you'. You probably would not want to say this if you wanted to meet friends that you see regularly to hang out together.
寂しい translates literally as 'lonely' although in this context can be understand as'I feel lonely', you might use this to emphasises your feelings more, if you are really missing someone close to you.
There is also 恋しい which is an adjective used for places or things but cannot be used for people.
Japanese Ammo has a helpful guide on how to say 'I miss you'.
Here are some example sentences:
I want to meet her
I miss my hometown
I will be lonely
Depends on the situation.
寂しい is to openly admit that you are also lonely.
We don't say such a thing as often as American people say 'I miss you'.
寂しい just means lonely; you can miss some specific person without being lonely in general.
Often, the meaning of "I miss you" is expressed with 会いたい. It appears in songs quite a lot (君を会いたくて、会いたくて). But of course that is literally about wanting to get together, not about the feeling of that person lacking, which are not exactly the same.
The problem is that sometimes you want to meet someone without missing them and that is also covered by 会いたい. Maybe you want to meet someone famous, whom you've never met; in English you can't say that you miss that person.
There are more direct ways to say "I miss": the verbs 恋しがる (がる form of 恋しい), and 懐かしむ.