If the spell required you to share a language with the creature, it would say so.
Let's look at something like Modify Memory (PHB, page 261)
You must speak to the target to describe how it's memories are affected, and it must be able to understand your language for the modified memories to take root.
It's specifically called out that the target must be able to understand your language.
Command (PHB, page 223) is another example:
The spell has no effect if the target is undead, if it doesn’t understand your language, or if your command is directly harmful to it.
Since there is no mention of language in Conjure X's description, it will follow your commands regardless of the language you speak them in (although I think it would be a reasonable assumption that it has to be a known language within the DnD universe, not just one you made up on the spot).
Note though that the spell only says
they obey any verbal commands that you issue to them.
It doesn't allow you to automatically understand the creature's language, give the creature the ability to actually understand your language (unlike the Find Steed spell), or give it any special ability to communicate with you.
As per @SevenSidedDie's comment, they may not even understand the actual commands, being driven more so by magical impulse at the intent behind your words rather than by the words themselves.
Most creatures are summoned unwillingly, or are otherwise unhappy about being summoned.
The Monster Manual is rife with examples of creatures that hate being summoned, under almost any circumstances. For example:
Devils: MM (67):
However it is summoned, a devil brought to the Material Plane typically resents being pressed into service. However, the devil seizes every opportunity to corrupt its summoner so that the summoner's soul ends up in the Nine Hells. Only imps are truly content to be summoned, and they easily commit to serving a summoner as a familiar, but they still do their utmost to corrupt those who summon them.
Yugoloths (a specific circumstance) (MM 311):
A yugoloth summoned using its true name, as inscribed in the Books of Keeping, is forced to serve its summoner obediently. The yugoloth hates being controlled in this manner and isn't shy about making its displeasure known.
Elementals (MM 123):
Certain spells and magic items can conjure an elemental, summoning it from the Inner Planes to the Material Plane. Elementals instinctively resent being pulled from their native planes and bound into service. A creature that summons an elemental must assert force of will to control it.
Demons (MM 53):
A mortal who learns a demon's true name can use powerful summoning magic to call the demon from the Abyss and exercise some measure of control over it. However, most demons brought to the Material Plane in this manner do everything in their power to wreak havoc or sow discord and strife.
(MM 51):
If a single mistake is made, a demon that breaks free shows no mercy as it makes its summoner the first victim of its wrath.
One exception is a Glabrezu (MM 53):
A glabrezu takes great pleasure in destroying mortals through temptation, and these creatures are among the few demons to offer their service to creatures foolish enough to summon them.
The Gate spell is explicit about simply pulling any creature away without notice:
When you cast this spell, you can speak the name of a specific creature (a pseudonym, title, or nickname doesn't work). If that creature is on a plane other than the one you are on, the portal opens in the named creature's immediate vicinity and draws the creature through it to the nearest unoccupied space on your side of the portal. You gain no Special power over the creature, and it is free to act as the DM deems appropriate. It might leave, Attack you, or help you.
The Conjure Fey spell implies that the fey resents being summoned, since it attacks you if you lose concentration (much like a demon):
If your Concentration is broken, the fey creature doesn't disappear. Instead, you lose control of the fey creature, it becomes hostile toward you and your companions, and it might Attack.
Confusingly, the Conjure Woodland Beings spell does not have that clause. Indeed, you could summon the same creature with either spell, and it will only attack you when you lose concentration only if you used the higher level version.
The only case that might allow for some free will is Conjure Celestial, because it has the freedom to only obey commands that follow its alignment.
It obeys any verbal commands that you issue to it (no action required by you), as long as they don't violate its alignment.
Best Answer
Summoning demons might be more useful than conjuring animals depending on the circumstances and desired results. For instance...
Demons are immune to poison
One trait that all demons have in common is that they are immune to poison damage and the poisoned condition. If you are fighting against enemies which rely on poison damage and status effects, they will have a harder time against a bunch of demons than against a bunch of mundane beasts.
Demons are inherently scarier and do weird things
For the common person, hostile wild animals are certainly scary, but hostile demons are undoubtedly more so; and where animals will simply physically attack their foes, demons tend to have extra magical or unusual abilities which make them more versatile. If your goal is to create as much fear and chaos as possible, a squad of demons will do that job more effectively than a gaggle of beasts, even if, according to game statistics, they're actually equivalently dangerous. Where town guards might be confident enough to take on a boar or bear, they might flee in terror at the sight of a demon... even lesser ones.
It also provides the DM a convenient character option for villains to use, since summoning literal demons is a good way to paint an antagonist as obviously evil.