The first Diablo is a very, very great game, but it has small, minor flaws that detract from the experience. Hellfire is an expansion that seeks to repair these flaws, and more. It also appends the main game with additional levels and a side-story involving some of Diablo's former minions.
The new content notwithstanding, much of the other contributions of this expansion include fixes and refinements which while mostly minor, were very much welcomed. Characters can now move faster around the town hub, Diablo gains a much deserved boost in power, there is a new spell for quick traveling within dungeons and there is another that highlights items on the ground (though this seemed to be little more than a disguised experimental piece of code).
The new levels and quests can only be accessed after reaching a certain point in the main, original dungeon - usually around the eight or ninth level. While this is a bit frustrating, this artificial delay is vindicated when the player realizes that the monsters who populate these levels are new breeds entirely; while they behave similarly to those in the main dungeons, they are more dangerous than those found in the first few levels of the latter dungeons.
The additional story offered by the expansion does not seem to be well-woven into the main story though. Perhaps Diablo keeps to the adage of "keeping your enemies closer", but having a majorly treacherous over-minion of his trapped away so close to the town of Tristram, which is essential to his plans, was quite an odd decision to understand. After all, said fiend is very, very angry, and the player has been forewarned that he would likely take out his frustration on Tristram first.
Of course, a wise player would realize that it is just a convenient plot device to justify the opening of another couple of new dungeons to raid for challenge and loot.
Speaking of loot, there is not much unique loot (akin to the Butcher's Cleaver in the main game) to be obtained from these new dungeons, an absence that is likely due to concerns about integrating them into the main game without bugs.
There are new expendable items to be had anyway, such as oils that can be applied to weapons to temporarily alter their properties and runes that can be used like traps. There are also new types of shrines, new weapon & armor grades and new spells (as well as new spellbooks in which to store them). These are fortunately well integrated into the game, though they only appear in this version of the Diablo game, not the vanilla one.
The new class, the Monk, appears to be a compromise between the Warrior and the Mage, though his tremendously increased attack speed with staves meant that he is best used against enemies up close and personal, together with spells for crowd control. He is a well-animated character, with certain melee attack animations allowing him to hit multiple enemies at once, which makes him even more impressive.
Unfortunately, much of Hellfire cannot be played together with the vanilla version. Perhaps creating the impression that this expansion could be experimental or even cut designs for the original game, the expansion does not appear to have Battle.net support. In fact, the new levels have to be played in single-player. The Monk, and characters made using this version of Diablo - or converted over - cannot be used not just in Battle.net, but local multiplayer games of the vanilla version as well (though the latter case can be overcome with third-party modifications).
It could be a consequence of Blizzard sub-contracting off the development work to Synergistic Software while it was working on Diablo II, but that would be just conjecture.
In conclusion, while Hellfire offered more to add to the Diablo IP, which is good, it does come off as not fully incorporated into the rest of the game. Even so, it does provide much helpful refinements to Diablo and enrichment of its single-player experience.