Yes, towards the end of the Amiga's life in the 90's the PC did eventually catch up and overtake it. The continued development of graphics cards and the falling prices of ram were key factors in that happening and the PC being easy to expand another. But you still had to pay over four times the price of an A1200 to purchase a gaming PC in those days.

The big change came with the advent of 3D. The Amiga was originally designed with 2D Sprites in mind and not 3D polygons and the way it draws the display isn't suited to 3D either. In contrast the PC graphics cards were starting to gain their own graphics memory and draw graphics in a way very suited to 3D. This meant that games heavily reliant on 3D such as Doom, X-Wing or Wing Commander were quite easy to code on the PC and they looked good. The Amiga was different enough to make 3D games much harder (but not impossible) to code.

The blame of the Amiga not keeping up with the PC is completely Commodore's fault. They sat back and enjoyed the success of the A500 for too long. The AGA chipset was developed long before it was actually used in the A4000 and A1200, and by the time the A1200 was launched in 1992 the successor to that chipset, the AAA chipset, was already developed and finished. That shows how slowly Commodore were working on development of new systems for market. By the time the A1200 was released the rest of the market had already started to catch up and it was only a couple more years before it overtook and started to leave the Amiga behind.

It just goes to show that when a company is controlled and run by a business man with profit margins and no actual interest in the technology that it doesn't work. At least Bill Gates was a genius programmer with insight and knowledge of the technology and it's development and progression. He may have had some wild ideas along the way such as 640K being all anyone would need (definitely a programmers pov, not taking graphics into account), or the Internet being just a "fad" and would "never take off", but you can't fault the guy for timing within the marketplace. Something Commodore never learnt later in it's life.