Most drip pots do not achieve a high enough temperature of water when introducing the water to the grounds. Water temperature is generally agreed to be best when in the 195 - 205 degrees Farenheit range for proper extraction. Also, you cannot control immersion times with drip pots. As previously stated, paper filters "trap" many of the coffee oils and they are unable to reach the coffee pot. In all, drip pots tend to brew a more static cup of coffee. A "cleaner" cup, devoid of many of the coffee beans' true flavors and nuances.
Press pots ("French Presses") allow exact control of the brewing. Exact water temperature can be introduced into the chamber and exact timing of its immersion (contact with the beans) can be exercised. The wire mesh screens they employ are only designed to keep large bits of coffee from the resulting liquid. A bit of "sludge" does come through. If you are unaccustomed to press pots, you may find this "chewier" cup unfavorable. Or you may learn to enjoy it for its less "clean", but more "honest" cup of coffee that is truer to the bean's taste profile.
I do use a drip pot as well. I rarely have the time to press pot. For my drip pot set up, I employ a Technivorm MoccaMaster drip pot with a SwissGold #4 coffee filter. SwissGold filters are a very good alternative over paper filters, in my opinion. They allow oils to escape through into the coffee. They also let through some of the "sludge" associated with press pots. A relatively pricier setup, but the durability of the machinery and the taste it yields is worth it to me.
