What is the difference, if there is, between this two distillation methods?
There is no fundamental difference between the two methods. 
In both cases steam is formed and it is passed through the plant material. 
The intention is to extract the non volatile water insoluble essential oils from the plant. 
The method is called steam distillation. 

On the other hand, normally it is not done in this way. 
Since generally only the essential oil is of interest, the integrity of the plant does not matter, it is placed in water and the whole is heated and the water is distilled. 
The students do this in the lab with cinnamon and orange peels, for instance. 
It is much easier because you only have to heat one vessel. 
In these two step setups, you must isolate the part containing the plant very well so that the steam does not condense, once the whole system is hot enough. 
Preferably, you should also heat the part containing the plant, especially in your second example.