A major effect of being isolated from other Civs is You have no means of monitoring or (crucially) influencing their progress.
"If you know the enemy and know yourself, you need not fear the result of a hundred battles. If you know yourself but not the enemy, for every victory gained you will also suffer a defeat" - Sun Tzu
Civilization is a strategy game. Your goal is (or should be) to defeat the other civilizations by outperforming them significantly in at least one of a number of ways (military, science, culture, etc.): how can you have such a strategy when you are crucially lacking both knowledge of your enemy and influence over them?
For example, noticing your isolated start and picking up a bit of luck with a handful of science wonders you decide to pursue a science victory. You settle a minor spread of cities, start churning out science buildings... what you don't know is that on the next continent Montezuma has killed off both his neighbours, annexed their territory and has built up a near-unstoppable army which can be readily refreshed from any of numerous production powerhouses. Whoops, he's found you! And in the next hundred years has declared war on you and if not steamrolled you entirely has at least been enough of a frustrating distraction to make Alpha Centurai a distant pipe dream for another thousand years.
Rolling the clock back... let's say you're one of Monty's hapless neighbours. Foregone conclusion, right? He's sure to dominate just as before. Well, as an experienced player you're able to anticipate the Aztecs' early aggression - you can monitor and anticipate their behaviour and adjust your strategy accordingly. You recognise that the Aztecs are a strong early-game civ that rely on quick contest to get the upper hand. You decide that during their peak years you'll prioritise defensive capabilities - more military, walls, founding fortress cities in easily protected & reinforced areas.
But you're also not alone! Another civilization is nearby and recognises the threat Montezuma poses. Far from being a direct competitor, you identify an opportunity to leverage them as an ally to flank the Aztecs. A bit of bribery and some kind words and they're happy to fight alongside you to your mutual benefit.
Time passes, the Aztecs pass their prime. They've been hammering fruitlessly against your walls for decades now and you've meanwhile been creeping up the tech tree... you press the advantage when the time is right, obliterating their laughably obsolete hordes with advanced troops, highly experienced from their years of brushing off invaders. You capture two, three, four cities... The Aztecs grovel for peace and you graciously permit them to live, though now a pitiful and backwards tribe; hardly the curbstomping empire they were last game.
At this point the game is cracked open: you could wall yourself in and start pursuing that science victory again, having effectively doubled your output from the captured cities - or you might push further, turning your eyes to the fertile land of your former ally (now having outlived their usefulness)... you've not just fought off a potential threat but gained a significant advantage from it!
The fact is, conflict is inevitable in Civilization - the planet isn't big enough for everyone to get along nicely and sit safely inside their borders all game. Resources are limited, space is limited, sooner or later someone will try to take what isn't theirs and the victor will come out a significant cut above everyone else.
War is pretty much a necessity at some point at all but the lowest difficulty levels and being far apart from everyone else may give you breathing space for the early years but you'll be missing out on all sorts of opportunities to hinder your opponents and build off their failures.
Games are won far before a victory. If you're not already winning by endgame you've lost. Sometimes even the midgame is too late. Isolation is akin to joining a casino night and never playing a game - there's little chance of everyone else bankrupting themselves; someone out there is winning at least more than they're losing. The key is not to avoid risk but pick it up where the rewards are juicy.