Generally speaking, you don't get the full bonuses from the improvement. Although for most cases, that doesn't necessarily matter - the difference between settling on an iron mine and working another mined hill vs settling on the other hill and working the mined iron is not usually much different.
Now, there are some cases where it will matter - any bonus specifically for an "improved resource" won't count settled spots. So the water mill bonus, God of Craftsmen, Temple of Artemis, etc... don't apply if you settle on the resource. So if you plan to make use of them, then I would tend to settle away from the resource. Also like you don't get the eureka for an iron mine for settling on it, only if you mine it. The resource will count towards adjacency (ie. an industrial zone will get +1 for being next to a resource if you build it next to the city centre), but does not get the improvement adjacency (the industrial zone's +1 for every 2 mines).
But as mentioned, the big benefit is that you get the bonuses earlier, and you free up another tile for later. So if you settle on that Tea tile for your opening city, you get the amenity and can sell it without unlocked the tech for it. Also early on especially, if the tile gives you a non-standard bonus and you would otherwise not want to work the tile until later, it can be a big boost early to settle on it. So the Tea, for example, if you settle on it your city gets the +1 science from the tile, and you can work other high food or production tiles instead. If you don't settle on it, my usual early game route tends to focus on those other tiles, so I often wouldn't want to work the tea until my city was up to size 6 or 7. That's even bigger for something like desert incense, where I really don't want to work that tile basically ever unless if if I get Nazca Lines or Petra.