What are the disadvantages of having an old, worn cogset?
As a chain wears out, the distance between links gets bigger (this is what chain wear tools measure). As this is happening, the chain will grind the cogs to match the worn chain (distance between teeth increases). This is why your gears might not feel so bad, but then they get much worse when you put on a new chain. The new chain's links do not line up with the teeth on the cogs. This will be noticeable in a few ways depending on how worn the cogs are. 

First is the noise coming the drivetrain and no amount of lube will silence it. The next one will be shifting performance as you find the chain getting stuck on a cog more often. If the cogs are extremely worn, you'll then get slippage as the chain slides forward over the cog (staying on the same cog,  not shifting) when you put pressure on the pedals. This last one can actually be a safety hazard as you might fall if you are out of the saddle pushing, then suddenly the drivetrain lets go (it feels similar to a broken chain when it happens).

Based on this, you might think that it's a good idea to keep the worn chain on for as long as possible, but that will lead to much faster wear on other components as well. Chainrings and jockey wheels rarely need replacing on a well-maintained drivetrain, but if the chain is too worn, then it will grind into these just like the cogs, making it a much more expensive replacement. Many people replace their cassette and chain at the same time once the chain length is 1% longer than when new. This ensures that you won't put excessive wear on the chainrings and jockey wheels. 

I prefer to replace my chain at 0.75% which allows me to keep the cassette for about 5-6 chains. This is cheaper for me since I use Dura Ace and Ultegra cassettes, but if you're using more entry level gear, then it's usuallly cheaper to wear them out together and replace them both at once.