In standard smithing, in order to reforge a blade like that, it is more than just "putting them side-by-side and doing one's best to seal them together." Typically, you heat up the end of the shard until the metal is softened, then take the shard that is cool and still hard, and insert it into the softened metal. Then you heat up the seam, and subject it to tempering: ye olde blacksmith beating on it with a hammer. That produces a fairly sturdy join, but the seam is always visible, and the sword is never as sturdy as it was originally. However, this was not standard smithing. This was elven smithing.
From The Silmarillion, "Of The Rings of Power and the Third Age":
The host of Gil-galad and Elendil had the victory, for the might of
the Elves was still great in those days, and the Númenóreans were
strong and tall, and terrible in their wrath. Against Aeglos the spear
of Gil-galad none could stand; and the sword of Elendil filled Orcs
and Men with fear, for it shone with the light of the sun and of the
moon, and it was named Narsil...
But at the last the siege was so strait that Sauron himself came forth; and he wrestled with Gil-galad and Elendil, and they both were
slain, and the sword of Elendil broke under him as he fell. But Sauron
also was thrown down, and with the hilt-shard of Narsil Isildur cut
the Ruling Ring from the hand of Sauron and took it for his own...
Thus Narsil came in due time to the hand of Valandil, Isildur's heir,
in Imladris; but the blade was broken and its light was extinguished,
and it was not forged anew. And Master Elrond foretold that this would
not be done until the Ruling Ring should be found again and Sauron
should return; but the hope of Elves and Men was that these things
might never come to pass.
From The Fellowship of The Ring, Chap. 3, "The Ring Goes South":
The Sword of Elendil was forged anew by Elvish smiths, and on its
blade was traced a device of seven stars set between the crescent Moon
and the rayed Sun, and about them was written many runes; for Aragorn
son of Arathorn was going to war upon the marches of Mordor. Very
bright was that sword when it was made whole again; the light of the
sun shone redly in it, and the light of the moon shone cold, and its
edge was hard and keen. And Aragorn gave it a new name and called it
Andúril, Flame of the West.
We are never told how the elves reforged the sword; they may have completely remelted it, and reforged a new blade; they may have done something similar to how a human would have done it (that's how Jackson shows it in the Return of the King extended edition). We do know from the description that it was a mighty blade, with the magical properties of sunlight and moonlight.
From The Two Towers, Chap. 7, "Helm's Deep":
Charging from the side, they hurled themselves upon the wild men.
Andúril rose and fell, gleaming with white fire. A shout went up from
wall and tower: "Andúril! Andúril goes to war. The Blade that was
Broken shines again!" ...Three times Aragorn and Éomer rallied them,
and three times Andúril flamed in a desperate charge that drove the
enemy from the wall.
This would seem to indicate that it has not lost any of its power from the First Age; we can only assume that is because it was reforged by the elves. Let's not forget that one of their smiths forged the Three. They were definitely talented in that area, possibly even in a magical way. From the descriptions given, it would seem that it was not any weaker or stronger, but the same power.
