I've started a few days ago this game but I never played Hellfire. What's the difference between them? In which way I have to play?

In single player, Hellfire will give you 2 new areas, each with 4 levels [plus new spells, enemies (only available in the new areas) and items].

You can start them any time after finishing the Catacombs (level 8), by talking to the Farmer. They are completely optional, meaning you can ignore them completely.

One of the major differences: In Hellfire, you can walk twice faster in town. This makes the frequent and necessary visits in Tristram a breeze. A very needed improvement. You'll also be healed instantly by Pepin the healer without having to ask him for it.

Here you can read about the many addition to Diablo with Hellfire, they are all worthy or even necessary:
diablo.fandom.com/wiki/Diablo:_Hellfire

If you play single player, there is no reason not to start a game in Hellfire.

I must disagree here. Hellfire is based off Diablo v1.04, and there are useful fixes in later versions of Diablo that were never merged back to Hellfire. Perhaps most notably, the logic for Mana Shield is incorrect in Hellfire and higher levels of Mana Shield are less effective than lower levels. More generally, any list of Diablo bugs fixed post-v1.04 will indirectly show you which bugs Hellfire still has. Pulling from The DSF Buglist for Diablo v1.09, I see (after removing the less concerning ones):

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VERSION 1.07

Fixed crashes associated with being attacked or killed by a Black Death (Zombie class). This includes the crash upon resurrecting after being killed by a Black Death.
Fixed the freeze upon touching a Hidden Shrine while equipped with only indestructible items.
Fixed the bug that prevented Rogues from properly disarming traps on levels 13 through 15.
Fixed bug that rarely transformed unique items (like the Optic Amulet) into some other (usually mundane) item when saving the game and then starting a new game.
Prevented repeated castings (128 or more) of Mana Shield from disabling the casting of any more spells.
Fixed bugs that prevented items with the suffixes Thieves, Balance/Stability/Harmony, and Piercing/Bashing/Puncturing from working properly.
Fixed the bug that caused Mana Shield to become less effective with increasing spell level.
Fixed the bug that caused excessive damage when a Mana Shield expired due to an attack.
Prevented bows with fire-hit damage from interfering with the behavior of Gargoyle class monsters.
Allowed right-clicking (casting/drinking) belt items in positions 5 through 8, while the Spellbook is open.
Fixed the bug that froze Lazarus (and made him invulnerable to arrows and spells) after player had entered his lair, then left (or died), and finally returned.

VERSION 1.05

Fixed potential memory corruption bug that occurred when casting town portal

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The freeze on use of a Hidden Shrine is easy enough to avoid if you know about it, but very frustrating if you trigger it, since you must close the program and revert to your last saved game.

The freeze of Lazarus could be game-breaking if there is no way to reset him and you cannot kill him, since you need to defeat him to proceed. The phrasing suggests you might be able to kill him with melee, assuming you have enough To-Hit and Damage to overcome his regeneration.

The 128-or-more Mana Shield bug sounds silly, but I'm pretty sure it has at least one, and possibly several, important consequences. It probably existed because you could have multiple Mana Shields up at once, and in conjunction with the bug for excess damage on Shield expiration, you could potentially take the excess damage for each excess Shield. Additionally, every excess Shield consumed a slot for an active spell, so even if you did not have 128 Shields up, you were still losing at least some slots for spell effects, which means Chain Lightning would start to fail even sooner than usual (and due to that same cap, Chain Lightning at high levels almost never works as well as it should).