Mainly more points of failure, and additional weight to my knowledge.
The electric motor allows superior city driving due to the nice fact that you have regenerative breaking and much more efficient low speed operation where an ICE would not be able to run at its most efficient RPMs. While the gas engine gives a lot of convivence that you might otherwise lose.
The downside is now you have two drivetrains that need to integrate into a very limited space, with limited weight as the whole goal was to make an more efficient car. Its also non-trivial to make a good transmission let alone one that has two inputs, or even an integrated electric motor.
Batteries in a hybrid are also pushed much harder in a way. Batteries are often rated over 20hr discharge cycles as you can generally get more power out of a battery by discharging it slowly. Asking 1A from a 100Ahr battery vs 1A from a 1A battery is very different in terms of effective capacity as well as the longevity of the cell. Given that batteries are so heavy and you need two drivetrains you can only add so much battery capacity. The end result is that its easy for the batteries to age faster in a hybrid.
A surmountable issue is heat, batteries don't play well outside a range of limited range of temperatures. Too cold and they will have less effective capacity, too hot and they can be damaged or degrade faster. This issue can be overcome, but is just another way there is more complexity than Tesla's 'the floor is a battery approach'.
Hybrids are cool, but they are hard to make right is the tldr.