Why did Amanto ban the use of swords at all when they had already defeated the Samurai with swords? What threat do they have even if the Samurais have their swords since they are already stronger than them (Samurais)? If they allow wooden swords which are also destructive when used by the existing Samurais, then how would sword ban do any good? The anime only says 'sword ban' was enforced but no plausible explanation was given as to why.
The Sword Ban was enacted because they feared of the Samurai's ferociousness. During the Amanto War, although the Samurai's main weapon is sword, they managed to deal massive damage, especially the group lead by Sakata "Shiroyasha" Gintoki, Katsura "Runaway" Kotarou, Takasugi Shinsuke, Sakamoto "The Dragon of Katsurahama" Tatsuma. The main reason behind the Sword Ban is to prevent heavy casualties on the Amanto's side.
It also has to do with the way the Samurai wage war. They do a guerrilla warfare, making complete obliteration hard to achieve. The current Jouishishi also performed their terrorism in cities populated with both Earthling and Amanto where the use of heavy artillery is out of option. While guns and artilleries have both range and power, they are less useful than a sword in a close combat, especially when used by skilled fighter.
Thus, we have the sword ban to ensure peace to a certain degree. They even created the bakufu's lapdog, the Shinsengumi and the Mimawarigumi, compromised of former samurais to ensure that the said samurais won't turn into another Joui factions.
Edit: About the Wooden Sword, it was also prohibited. The Cheetah/Leopard/whatever it was amanto in episode 3 of the anime asked Gin who the hell he was for defying the sword ban.
Edit2: Most people in Gintama uses steel swords (see the Shinsengumi, Yagyuu Kyuubei, the Jouishishi, etc). The only one that uses wooden sword in real combat is Gin-chan. Why do Gin-chan uses wooden sword? If you try asking this to him, he'll probably answer with it makes him looking more badass and more recognizable, which is important since he is the main-protagonist. You can see the episodes about Gintaman to confirm this. See the "n" there. It is not a typo. There is the episode where Gin-chan meets the editor or JUMP in a train and then later helps him devoping Gintaman. Besides, he bought the sword from TV shopping anyway.