Hi Eric,
There is a difference. "Al lado de" means "beside, next to" and speaks principally of where something is physically in relation to something or someone else. When used metaphorically, like being beside someone through an experience, I think it may be more common to say something like "estoy a su lado."
"Junto a" literally means "joined with" and can also be translated as "with." Something can be "al lado" and not "junto a". Or they can otherwise both be the case. It can also describe a non-spatial relationship between two entities. If someone said "Yo estoy junto a mi esposa ya hace 5 años", that would mean he was with his wife for 5 years.
So, the difference between the two expressions are not matter of elegance.
Hope that helps.