There are third parties that try to provide OS emulators. Apple is no help. They are a considerable hindrance.
Does Microsoft do better? Having been a long time Mac users, I don't know.
Classic will still run under OS X.4 Tiger, but not on Intel Macs, only PowerPC Macs. It is not included with the distribution of Tiger but it will work with the install from the previous version of OS X.
There is no reason that Apple has to maintain support for a ten year old, obsolete version of the OS. They have bent over backwards to maintain compatibility for six years beyond the introduction of its replacement. That should have been plenty of time for you to upgrade your apps and convert your files. You had plenty of lead time and warning.
Microsoft tries to maintain backwards compatibility and it is a major problem in letting them upgrade their OS. Too much legacy stuff must be retained for efficient improvements. So yes, they do a better job of backwards compatibility. . . but that may be more of a curse than a blessing because it gets in the way of moving toward the future of computing.
Classic works well under OS X.4 Tiger but only on PowerPC Macs, not Intel. It does not come with Tiger but will work with the already installed version of Classic from Panther. It was removed completely with Leopard. Apple is not responsible for the writers of BookUp's chess programs for not upgrading their product. Apple has made every effort to make it easy to upgrade.
Microsoft makes backwards compatibility one of the primary sales points of their newer OSes... but that may be more of a curse than a blessing as the complexity of maintaining legacy code gets in the way of designing a modern OS. So, yes, Microsoft does do legacy code better... but a lot of older code does not work in any case.
It looks as if you will have to keep an antique Mac around for those legacy apps. Sorry about that.
Check out SheepShaver, a Power PC emulator that runs on new Macs, including intel machines. It allows you to then install and run earlier versions of the Mac OS (7.5 thru 9).
With regard to backward compatibility, a whole lot better. There is darn little, going all the way back to MS-DOS, that won't run on the current version of Windows.
Now, Apple probably didn't plan on obsolescence by design, but decisions they made at critical times have sort of forced their hand. There's a lot to like about the current Apple OS, especially if you're working in those markets to which Apple has traditionally catered, but their hardware is no longer anything special.
In my opinion, it is inevitable that erosion of the distinctiveness of the OS itself will follow. Indeed, it may have already begun but they've been riding on Vista's failure to deliver for the last couple years, and if Seven lives up to the hype they are about to face a challenging time.