Both the elves and the dwarves were famous for their metal work in the Lord of the Rings trilogy, but what is the difference between what they made, and which one had the better skill of making amours and swords?
James Christopher's answer sums up the second part of your question well, but as to the difference in what they made, a little more detail is needed.
Once the Elves learned to forge with steel,
the shape of the sword changed, now being able to take on the form of a great broadsword or a light and agile curved sword. Additionally, they took great pride in decorating their swords. As we see in the Lord of the Rings, some swords like Sting had magical properties such as glowing blue when orcs are near. As far as the use of Mithril, lotr.wikia has two contradictory passages:
Thus, Elven blades became renowned as great weapons, capable of performing deeds beyond the skill of their handlers and were even more glorious when the use of Mithril was allowed to the Elves.
Thus implying that Mithril was used on weapons and,
The Elves loved it for its beauty and presumably used it for jewelry and attire rather than weapons or armour.
Implying that it was used for decorations only. Neither entry is sourced, however.
The Dwarves used both swords and axes, with the axes being the more ubiquitous weapon. The axes certainly benefited from the use of Mithril, as the lotr.wikia notes:
With Mithril, Dwarves were able to reach their peak in making weapons. Suddenly, axes began taking on almost magical qualities, being able to slice through almost any metal. The metal Mithril may have completely changed the way Dwarves made and used their axes, but the magical metal was hard to find and even harder to harvest. Thus, most Dwarven axes continued to be made of other metals like steel.
The sword likewise benefited from the use of Mithril,
As with the axe, the Dwarven sword reached its peak when the use of Mithril came about, but also like the axe, it still was primarily made out of steel.
Both races were skilled in the craftsmanship of their weapons, but the dwarves having the earlier edge in materials, seems to be the reason they were regarded as the superior smiths. The swords of both races used Mithril sometimes, and both would at times contain magical properties.
The quote provided in James Christopher's answer gives an overarching view of this. This superiority was likely due in part to the dwarves' use of Mithril. As it was said of Frodo's Mithril-coat, one was
worth more than the value of the Shire and everything in it. But the loss of Moria, the greatest, if not the only Mithril mine, put a damper of Mithril armor production. But they were not just superior in their materials, but the forging process itself used by the dwarves was superior, giving them an across-the-board advantage.
According to lotr.wikia,
not much is said regarding Elf armour only that it is bright and beautiful to look upon...However, much of Elven armour is derived from Dwarf armour, for the two races have great influence upon one another.
The films portray elves as wearing some sort of plate armor, but the truth is, we don't really know what they used. It is unlikely they used Mithril mail, so whatever they had, it was less effective than dwarven Mithril armor. Since the steel armor of the dwarves was superior to elven steel, it seems reasonable to assume that elven armor, regardless of its composition or design, was less effective than standard dwarven armor.
Tolkien writes: "In the tempering of steel alone of all crafts the Dwarves were never outmatched even by the Noldor, and in the making of mail [...] their work had no rival."