The old paper speakers will be more bassy because they are a lower rated speaker and you have jack all power behind them. The newer speakers you bought require more power to give the same volume. An amp will fix this and you will be able to give them the rms they require to make bass.
I had a 6" pair of speakers from my old mazda (stockers) that I transported to another car because they were so efficient. They were only rated at I think 10 or 15w but sounded great when mounted well and behind a reasonable head unit.
Get a cheapish amp and see how you go.
With an amp you can adjust the gain. At medium level on your head unit you may be abl;e to dish out 30w rms or more dependingon where you set the gain, where your head unit would only give out say 5w rms