Does the yellow stick smash have less impact than a normal joystick smash in Melee?

According to the super smash bros wiki, the C-Stick is just a shortcut and changes nothing to the damage done or the attack's range :

C-Stick

[...]

In Brawl, the C-Stick (And right stick on the classic controller) can be assigned other control behaviors:

Smash: (Default) Smash attack short cuts, just like in Melee

In other words, when you're "C-sticking", the smash damage is exactly the same as the using the combination of left stick + A, though the C-stick can be executed much quicker.

Additional interesting information can be found about the advantages and disadvantages of both methods:

The C stick is one frame slower than using A and cannot be charged. The A, on the other hand, can do more knockback (with the charge) and is easier to repeatedly tap. In short, the C stick is just like using X to jump or Y to grab. It's a different mechanism for control that can be used without any form of bad sportsmanship.

Recall that smash attacks can be charged. Thus a fast smash is also a weak smash.

A smash attack using the A button will almost always be more powerful than a C-stick smash; because the human reactions are slower the smash will be charged for at least some frames, while the C-stick smash has no charge at all.

This changes in SSB4, where the C-stick smash can be charged, but also becomes slower (in my hands anyway) than an A-button smash.