There are many studies comparing the effects of these two types of braces. Both self-ligating and conventional braces can correct any kind of dental problem that is suitable for treatment with braces. The most important condition for an excellent result is the treatment plan and how well it's carried out technically. Still, there are small differences between the two types of braces that have been demonstrated in scientific publications, rather than simply being claimed by the companies that sell them.
Self-ligating braces straighten crooked teeth faster than conventional ones. This doesn't shorten treatment overall, because the later phases of treatment progress more slowly, and in the end treatment takes just as long with self-ligating braces as with conventional ones. Self-ligating braces generally produce a broader smile as the final result, as their archwires allow more expansion in the side teeth. Before you rush to choose these braces, bear in mind that this expansion comes from the side teeth tilting outwards. In the final result, the contacts between the side teeth often don't look tight. Pain is said to be less with this type of braces. This is probably true, as self-ligating braces apply lighter forces to the teeth.
Conventional braces allow better control over the tilt of the side teeth, and with them we can finish the case with the upper premolars fitting snugly against the lower ones. They also come in colours, which motivates some children to wear them. In terms of price, self-ligating braces are more expensive than conventional ones.
This answer is for general information. Your own case needs an examination by an orthodontist.