Big Goombas first appeared in Giant Land in Super Mario Bros. 3, where they are not stronger than regular Goombas. In the New Super Mario Bros. subseries, when jumped on they split into smaller Goombas.
Their behavior is the same as that of a Goomba, and they share the charge as their sole method of attack. Unlike their smaller counterparts, they are much more resilient, being able to sustain multiple blows and jumps on their heads before being defeated. Giant Goombas run faster when charging, and as expected, its charge attack is much more powerful, causing high damage and knockback. Giant Goombas also have some flinch resistance to weaker attacks, which can make it difficult to stop a charging Giant Goomba.
Despite their high HP and power, Giant Goombas are still of relatively low threat. Having the same behavior as Goombas, their AI is rather basic, and Giant Goombas take much longer to begin their charge than Goombas do. When jumped on, Giant Goombas are also stunned for a sizable period of time while taking significant damage. Thus, Giant Goombas can be defeated with relative ease by simply jumping on their heads till they are defeated, which can easily prevent them from being able to begin their charge attack.
A giant-sized Goomba, as the name suggests. In Super Mario 64, punching and kicking had no effect on this brute. It could be stopped like normal Goombas- by jumping on it from above. A colossal Goomba appeared as the boss of world four in New Super Mario Bros., but that's a Mega Goomba, not a Giant Goomba.
Giant Goombas, now referred to as Big Goombas, appear as enemies, with mostly the same behavior as their Subspace Emissary version. They now have an attack where they stomp their feet down, which damages and trips the player. They can be attacked normally or just jumped on three times for a quick defeat. Despite their weakness, they drop above average power-ups.
It's a Goomba, only bigger! And much like normal Goombas, the worst they'll do is run at you. Unfortunately, their size actually makes this attack pretty dangerous! They can also stomp to knock you off your feet. They have a lot of health, but they take massive damage from being stomped on, just like a regular Goomba. Fortunate!
These giant-sized Goombas make quite an impression. Some will net you a coin when you jump on them, and others will split up into smaller Goombas. They're often much stronger than normal Goombas, but jumping on them a few times usually does the trick. There's also an even bigger one out there: the Megasparkle Goomba. He's so shiny!