The process of language learning, in general, is the same when babies learn their first language, as when adults learn a second language. However, the differences in brain function, amount of neurons, and affective filters play a role in whether the learning will be successful or whether it will be complicated in both cases.
Your premise asks to compare babies and adults and how these two subjects would respond to learning their first and a second language, respectively. Since we are not referring to a specific learning intervention, the first thing to consider is the state of mind of the learner.
In the case of babies, they are "clean slates". They have never been exposed to the sounds that would connect to a meaning. They cannot scaffold, that is, add on to information that they already know.
However, babies have the highest production of brain cells which, with a low affective filter (with no bias or fears of learning) would enable them to instantly create connections between sound and meaning. In all, a baby's process of first language learning is natural and instinctive. It is proportional to the immense amount of brain power that is developed at that age, the lack of emotional blockage that may prevent the learning, and the willingness for learning that comes out of the natural curiosity of babies.
Contrastingly, second language learning in adults is not an inherent, nor natural process that comes as easily. This is not because we cannot naturally process a second language, but because adults have learning barriers that may intervene with the process itself.
Before adults can learn a second language, they basically have to place themselves in the position of, you guessed it, babies! They have to clear any bias or fear of learning, activate their natural curiosity, motivate themselves, and be open minded- like babies are!
This being said, adults would learn quite fast by immersing themselves in the second language. This way, all the senses would become integrated in the learning, sending faster impulses to the brain. As a result, the adult mind would make quick and accurate connections of sound-meaning-symbol.
Therefore, it is the main task of any language instructor to immerse and motivate students and learners to engage in the learning process and enjoy it as naturally as possible.