Do power MOSFETs have significant disadvantages over normal MOSFETs?
Of course, they have some disadvantages, and in some applications, they are totally unsuitable. Both types of mosfets offer some compromises on different aspects.

First: regarding the price, I don't know where you got the estimations, but they're inaccurate. Mouser offers 2N7002 starting from about 0.12€, whereas IPS105N03L is about 0.68€. So it's not exactly cheaper. Maybe you were looking for a specific packaging of 2N7002 (TO-220, perhaps), which made the price much higher, but 2N7002 is actually one of the cheapest mosfet available.

Now, even without looking at the price, there are parameters other than ID, VDS and RDS(on) that characterizes a mosfet. For example:


VGS(th), the gate threshold voltage. For power mosfets, it is usually higher than small mosfets. Small mosfets can be driven directly with a 3.3V voltage, which is convenient. This is not always the case for power mosfets. However, for the two mosfets you compare, there is no significant difference.
The gate charge. Here, for IPS105N03L, we have 14nC of total gate charge, whereas for 2N7002, it is less than 1nC. Basically, this means that, when the mosfet changes state, the power mosfet will require a lot of current through its gate to overcome the miller effect. So you need big drivers to make it switch fast. However, the 2N7002 can be very easily driven, and even a simple MCU output pin can drive it as fast as you want. So it makes your design much simpler. This is where small mosfets shine.