I’ll assuming you’re talking about the cross sectional profile.
Different shapes offer differing levels of performance. Very generally speaking, rounded ‘O’ shapes are less stable and typically have lower spin times due to the distribution of weight. Without getting into all the physics of a spinning yoyo, more weight towards the rim = a longer and more powerful spin. This shape is less forgiving as there’s usually more surface area for the string to rub against, increasing the likelihood that the yoyo tilts or slows down; this is referred to as a ‘high wall’ and is seen more prominently in older yoyos as its more or less been phased out over the years for performance reasons. Examples of this shape include the OneDrop O-Shape Benchmark, A-RT Grail, and CLYW Peak 2.
On the opposite side of the spectrum, a hard ‘V’ shape tends to be found on more competition-worthy yoyos. The higher allocation of weight on the rim means the yoyo can spin harder and longer, thus decreasing the need for as many binds (which leaves more time for tricks, speaking in a competitive setting). Because of the ‘low wall’, the string rubs against the yoyo less, indirectly increasing the spin times, as well as making the yoyo more inclined for horizontal/change of plane tricks. Examples of this shape include the YYR Draupnir, YYJ Phenom, and OneDrop V-Shape Benchmark.
The other variations in shapes (W and H being the most notable) are in the middle, with the typical modern yoyo leaning more to the ‘V’ shape than ‘O’.
I personally prefer ‘W’ and rounded ‘V’ shapes as hard 'V’s usually aren’t too comfortable. It’s always nice to have an assorted collection, though.