An aftermarket amplifier can (or in most cases, will) be of better quality in general. An expensive 100W amp will give better sound quality than a cheap 100W one. Simply explained: at 100W, a cheap 300W amp will give quality comparable to an expensive 100W amplifier at 100W, its full power.
With all amplifiers the sound quality gets worse when you increase the volume. So a 45W stock radio in your car will only sound reasonable to say, 10W. With the more expensive, established brands it is less so. If you get a Bose, Sennheiser, or JBL amp, it'll still sound ear caressing at full power. However, your average Sony or Pioneer CD player will sound terrible at only half its power.
At 45W it will still make noise, but it'll sound terrible. It'll make cracking sounds where you expect goose bump inflicting bass. If you google for 'harmonic distortion' you'll find out how it works exactly.
Like you, i also wanted fair quality at a normal volume. If you're going for the established brands, you'd need a 15W amp for that. I chose a 100W amp from AliExpress, because at 15W that still gives good quality, and it is still a lot cheaper than a good amplifier.
In other words, if you're going for a cheap amp, you need to heavily compensate on the output power to keep good quality. It's also worth learning the difference between Class A,B,AB,C, and D amps. These classes tell something about the technique, not necessarily the quality.
In my experience, the price of the speakers won't make that of a difference. The only thing is that most cars only have a single wide-range speaker in each door, while it's better to have separate woofers, mid-ranges and tweeters.
Lastly, decibels and output power have a logarithmic relation. If you have an 15W amp of which you think: "i'd like to have twice this volume", then you'd need a 150W amp. Output power increases factor 10 per doubling of the volume.