Approaching Paris, the city's beautiful architecture came into focus
As FumbleFingers mentions in his comment, the first sentence is not grammatically "correct" because of the dangling participle. "Approaching Paris", does not modify "the city" but rather the unsaid subject "I", which (for some reason) is missing from the sentence.
Dangling participles are not unusual in colloquial writing where the subject can be omitted, if it's obvious, or to create a certain rhythm to the words. Creative writing has a broad license to break the usual rules. However, it's not good practice to use them in more formal writing, since they can be confusing and they can make your writing feel unfinished.
Again, as previously mentioned, if you need to get the answer correct on an exam you would rewrite it as:
Approaching Paris, I saw the city's beautiful architecture come into focus.
The second sentence is grammatically correct, but it's a different structure from the first sentence. The initial "finally" is an adverb that modifies "approached" but otherwise does not affect anything else in the sentence. and can be omitted without changing the essential information:
As I approached Paris, the city's beautiful architecture came into focus.
Which isn't to say it's not significant. "Finally" does add context by suggesting a long and/or difficult trip to Paris, and in a single word implies there is more to the story, although it might not be important at the moment.
To answer your question in your most recent comment: "Approaching Paris" is a participle phrase that modifies the subject "I". It describes what I was doing at that moment, but by itself it's an incomplete sentence. Other examples (from the above linked page):
Floating in the pool, I marveled at the clouds.
Biting his victim, the vampire felt a momentary thrill.
Beating you over the head with examples, I hope to make you understand participial phrases.