A trickle filter is technically more efficient because it allows more surface area in contact with the air, which raises the dissolved oxygen level in the water, and also exposes of the media to oxygen from the air, so there is more oxygen available to the biological bacteria (and also the fish, which is more of a worry for some species than others), which they must utilize to complete the nitrification process.
In submerged filtration the bacteria are able to utilize only the dissolved oxygen present in the water, which the fish are also consuming. If the filtration becomes clogged or something then there will be even less dissolved oxygen for the bacterial colonies, lessening their efficiency.
But IMO, the average hobbyist is going to be able to grow plenty of bacterial colonies whether they use submerged or unsubmerged media (whether it be trickle filters, biowheels, etc.). I have used many filtration types, both submerged and unsubmerged medias, and never had issues with ammonia or nitrites being present in the water, and some of my tanks have been VERY overstocked by most hobbyists standards.
A trickle filter may be more efficient, but the end product is still going to be nitrates, and in an extremely overstocked tank you will have nitrate issues that need to be taken care of with water changes, no matter how efficient your filtration is. So it's not really good to assume you can overstock just because you have something like a trickle filter which will more efficiently nitrify. Yes, you will be sure you have no nitrites or ammonia, but excess nitrates are a worry also.
No matter what filtration type you choose, I think the best bet is to add as much biological media as possible and pay attention to not overstocking the tank. Choose filtration that is convenient for you, compatible with your tank setup (for instance if you inject co2 for plants a trickle filter would not really be compatible since it would offgas the co2) is appealing aesthetically to you, etc..
For the average person, any decent filter, with plenty of media capacity, is going to work as long as enough biomedia is present and the tank is not too overstocked. If there is enough dissolved o2 for your fish to be happy (not including air breathers like gouramis, since they don't technically need dissolved o2) then there should be enough to grow sufficient bacterial colonies given enough biomedia.
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