First off - to specifically answer your question, yes, it is normal to purchase new speakers when you replace your factory head unit. Factory speakers tend to have lower power handling than typical aftermarket replacements. Smaller factory speakers often use “whizzer” cones for high frequency reproduction, as opposed to a separate high frequency transducer which will reproduce the upper range more accurately.
One thing you should be aware of is that power does not exhibit a linear relationship to performance of an audio system. In fact, a fifty watt-per-channel head unit can only be twice as loud as a five watt-per-channel unit. Many factory decks produce about six watts. If you double the power, you get a volume gain of only 3 dB. Doubling the volume requires a tenfold increase in output power.
What is the THD (total harmonic distortion) rating of your new head unit? This is a measure of how accurately the output from the unit represents the actual program material. Next thing to check, as has already been mentioned, is the wiring. Often, quick and dirty installations are accomplished by means of an adapter to your vehicle’s stock wiring harness - essentially connecting your new head unit to the existing wiring. A major drawback to this is that the existing wiring was designed with the stock deck in mind, and may not be sufficient for the increased current from the new unit. At high power levels, the resistance in the wire may be too high, which can manifest itself as distortion in the audio signal. Without exception, I have always heard better performance out of installations where all new wiring was run. Considering that wire is cheap, it makes sense to do this.
Next, the speakers could simply not have the power handling characteristics necessary to handle the output from your new head unit, although you mentioned that you hear the distortion at lower listening levels than with your old deck? In that case, what immediately springs to mind is that the system was tested after installation, and the volume was turned up to a point which exceeded the power handling of the factory speakers, damaging them in the process. One thing you may want to do is to remove and inspect your factory speakers and power connections. Realistically though, you do probably want to replace the factory speakers with higher quality drivers. Otherwise the increased fidelity gained by replacing the head unit is lost in reproduction.
Contrary to popular belief, when a speaker is “blown”, it is usually distortion that is the culprit and not simply the amount of output power. If you imagine your program material as being a waveform with peaks and valleys centered about a mean listening level, in almost every case the speaker should be able to reproduce that signal, provided that the amplifier (internal, in the case of your head unit) has sufficient headroom to accurately reproduce the peaks in the program material. Of course, these peaks may occur at amplitudes significantly higher than the mean listening level, necessitating a significant amount of power to reproduce accurately. As the mean listening level is increased, it becomes increasingly difficult for the amplifier to reproduce the extreme peaks without distortion. When this does occur, the amplifier “clips”, or chops off the peaks at an amplitude equal to the maximum amount of power that the amplifier can deliver. This is what you hear as audible distortion when you turn your music up loud, and what is likely to cause damage to your speakers if it happens at a level close to the speaker’s power handling ability. When using an amplifier with sufficient headroom to provide enough power to reproduce the program peaks accurately, it is very difficult to blow a speaker with undistorted clean program material as a result of power alone
So, I can’t say for sure what caused your problem, but at a guess, I suspect that the head unit was installed and tested at an elevated level, which sent a distorted signal to your speakers as a result of a combination of the wiring and poor accuracy from the amplifier output at the high power level, which blew your factory speakers, which will now sound crappy no matter what you send through them. Not a huge loss, as you probably want to replace the factory speakers anyway, but you need to be very careful with these car audio guys. As a general rule, they’re the sleazy used-car salesman applicants who were deemed to have insufficient ethics to sell entire cars.