A few years back, I spent some time making unofficial tests with an official LAN adapter, a no-name 3rd party LAN adapter (ordered from the Hong Kong-based DealExtreme website), and the built-in wireless adapter.
Tests involved timing how long it took to download various games from the Wii Shop Channel, pinging the console from other PCs, and playing Medal of Honor Heroes 2 online (looking only for general observations when playing online, since I had no quantifiable metric available).
The results* were surprising. Performance was nearly identical for both LAN adapters, and any differences observed appeared to be statistically insignificant. Using the wireless adapter was slower initially... but after tweaking my router configuration and location, that delay was reduced to negligible levels. The only real advantage I saw when comparing all three connection types was slightly more consistent results from the two LAN adapters. The most important factor from my tests was clearly the quality of service offered by your network provider, instead of anything related to the actual Wii console.
I chose to sell the Nintendo LAN adapter, use the built-in wireless adapter for regular play, and hold onto the 3rd party LAN adapter in case it will be useful in the future (ie: wireless router not working, a more consistent connection required for some online gaming, or visiting friends without wireless access). Some people might prefer to pay more for the official Nintendo LAN adapter because it probably has better quality assurance and resale value. Personally it wasn't worth the extra cost for me; you might have a different opinion about the matter though.
*Note that these tests were very rough in nature, and my background in networking and statistics is very limited. So "results" are better called observations and opinions.