Short answer: Yes
If you live a better lifestyle, you make powerful and influential friends. You can get a better lifestyle for free by being employed by an organization, or by training Profession or Survival.
If you want to train, research, or rest during downtime, you cannot work, and so Modest is not actually free anymore.
Modest is not the only free lifestyle
Page 187 of the PHB tells us this:
You can work between adventures, allowing you to maintain a modest lifestyle without having to pay 1 gp per day
and this:
While crafting, you can maintain a modest lifestyle
without having to pay 1 gp per day, or a comfortable
lifestyle at half the normal cost
So Modest is free only in the sense that you work to earn 1gp per day, and then spend that 1gp on your lifestyle. But you can get better jobs - the same section talks about being a professional:
If you are a member of an organization that can provide gainful employment, such as a temple or a thieves’ guild, you earn enough to support a comfortable lifestyle instead.
Still in the same section, you can also spend downtime as a rock star.
If you have proficiency in the Performance skill and put your performance skill to use during your downtime, you earn enough to support a wealthy lifestyle instead.
There is also a sidebar on page 159 about self-sufficient living:
Proficiency in the Survival skill lets you live at
the equivalent of a comfortable lifestyle.
So with a tiny bit of investment (Survival trained, or roleplaying as a guild member), a character can have a Comfortable lifestyle for "free". Just having Performance skill proficiency earns a Wealthy lifestyle.
Better lifestyles lead to better friends
Page 157 of the PHB says this outright:
Your lifestyle choice can have consequences. Maintaining a wealthy lifestyle might help you make contacts with the rich and powerful, though you run the
risk of attracting thieves. Likewise, living frugally might help you avoid criminals, but you are unlikely to make powerful connections.
Page 158 discusses lifestyles, both their costs and their benefits. The drawbacks are mostly along the lines of "you will get stabbed by hobos" while there aren't really benefits. But these entries also describe the kind of people you rub shoulders with:
Modest: ... Ordinary people living modest lifestyles include soldiers
with families, laborers, students, priests, hedge wizards,
and the like.
These are your peers, colleagues, and associates during downtime. They are not interesting or helpful to your life as an adventurer. Priests sound like they might be useful but according to page 187, being employed by an organization lets you live a Comfortable lifestyle. These are more like wandering preachers and acolytes-in-training rather than your friendly neighbourhood Cleric of Lathander that will cast heals on you.
Comfortable: ... You associate with merchants, skilled
tradespeople, and military officers.
It's hard to understate how much of a step up this is from Modest. You are now hanging out with people of means and influence. Merchants and skilled craftsmen are the best way to get your hands on better gear. Military officers are an amazing source of quests and leverage on the local government. These people are also all powerful, whether in the punching sense, or influential sense. They are good friends to have. Anyone belonging to an organization is at least in this tier, so if you want to make connections with resources, you're going to need to be here.
Wealthy: ... You live a lifestyle comparable to that of a
highly successful merchant, a favored servant of the
royalty, or the owner of a few small businesses.
This is a step up on both the means and the influence ladder. Living comparable to these people means you are in their social circles. Tell me who can get you sweet magic swords more reliably than a highly successful merchant, and who can convince the Duke to let your necromancer party member out of jail if not his trusted servant. Make friends with Wealthy people and you will go far... but if you yourself do not live this lifestyle, you'll never get invited to their parties.
Note that unlike Aristocratic, these don't have any drawbacks or threats to personal safety associated with them. You are still beneath the notice of political movers and shakers, but every lifestyle upgrade gets you closer to being able to benefit from their power.
You can't always work
We've covered good lifestyles. But why might a character want to live in poverty rather than comfort? Simple - he doesn't want to spend time working. Even Modest requires you to work for that 1gp per day. Practicing a profession is only one of the downtime activities listed on page 187.
Crafting gets you that 1gp as well. But characters can also be Recuperating, Researching, or Training - and the latter two options actually cost 1gp extra on top of your expenses! Especially poor characters who must nevertheless rest, train, or do research might be tempted to pick poorer accommodations.
Why do costs matter?
You're an adventurer, right? You wield sums of money that put entire nation-states to shame. Well...not really. At least not right away.
This page discusses how much money a 5e character can be expected to have.
After completing level 1, the party earns an average of 376gp. Split four ways between your murderhobos, that's a princely sum of...94gp. Even at a Modest lifestyle, that buys you three months of loafing around before you have to either go back to work or find another dungeon. In a world where level-appropriate challenges don't grow on trees, that's not as easy as you might think. That's why it's useful to hang out with high-status NPCs - they'll find you adventures, so you don't have to.
At level 3, the story is the same. At level 4, it's twice as much - a whopping 188gp per person. Less if you ever want to upgrade your equipment from the -5 rusty butter knife of shame that you started adventuring with. A modest upgrade to studded leather is 45gp, a whole month and a half of living expenses! Level 4 brings only a modest increase (282gp apiece). At this point your challenges are probably difficult enough that you want to buy some potions of healing at 50gp a pop, which leaves little spending coin. Or maybe a 75gp horse, since carrying literal tons of gold from the dungeon to home is a little back-breaking.
If you manage to complete level 5, you move up a tier of treasure hoards, and make it out alive with 3408gp. Shame that if you're a warrior, half of that goes to pay for your full plate (1500gp).
After this, you're coasting pretty well - even a year of an aristocratic lifestyle per delve is within your reach, provided you didn't want to save for a nice long retirement. There may be other incidentals that force you to consider finances - bribes, resurrection costs, etc. If you want to have naval adventures, seaworthy ships costs thousands of gold pieces.
If you do want to save for retirement, let's say you want to live at a Comfortable lifestyle for 30 years (much more if you're an elf). That's 11,000 days at 2gp a pop. Your adventuring spoils from the early levels won't get you there, so you may prefer working during downtime just to maintain your stash.
