When you're immersed in a language's country, you'll be exposed to not only the language, but also:
- The culture: an integral part of understanding the language; the language grew or morphed based on the culture.
- The language, as it's spoken in day-to-day life. Sure, maybe you can introduce yourself, you can describe where you live, what you need, where you want to go, etc., etc., but maybe you're not quite up to par with idioms, expressions, slang, informal speech, dialect(s), grammatical inconsistencies, bartering, jokes... everyday speech.
Immersion may also have unique effects on how the brain processes the language. From this study:
...measures of brain processing showed that different types of training led to different brain mechanisms.
"Only the immersion training led to full native-like brain processing of grammar... ...So if you learn a language you can come to use native language brain processes, but you may need immersion rather than classroom exposure."
According to this, immersion leads to native-like processing of a language (specifically grammar) better than a classic classroom approach.
As far as learning a language faster, I would say, yes, you would learn it faster, but only if you're still concentrating on learning. Living in country X is not going to magically convert you into a native speaker.