In some respects, higher difficulty levels CAN be easier:
1. Others have already noted that the AI has a lot more gold on higher levels, which you can trade for and/or exploit.
2. The AI also does better teching, which can help you by decreasing the amount of science required for techs discovered by other civs, giving you science bonuses from trade routes, and increasing the amount generated from Research Agreements.
3. If you aren't planning to bother with religion, then higher levels mean that the AI will spread his religion to you (and improve the religion) faster.
4. AI cities will be better developed, which makes them better if you conquer them. (Most relevant with civs that have an extremely strong UU which can compensate for the higher difficulty level).
Also, while higher difficulty games may be "harder" in terms of direct competitive with other civs, for Game of the Month and Hall of Fame games, they are actually "easier" in terms of finish time (due to the above factors).