The Chinese eggplant will have a thinner skin, less seeds (read: less bitterness); as a result, it will have an overall more delicate flavor.
You can substitute a globe eggplant (the kind usually sold in American grocery stores) equally; just be aware you'll get extra "eggplant" flavor.
Since the recipe you're using has strong flavors already, this should not be a problem. However, if the recipe were using more delicate flavors, it would not turn out quite as the recipe author intended.
In Britain (and France), the large purple varieties are known as aubergines. Other (pale and/or small) varieties aren't usually found outside of Asian supermarkets, where I imagine they are still referred to as brinjal.
The name 'eggplant' is used in the US, Canada, and the Antipodes, mainly because the lighter varieties are more common there, which arguably have the colour of eggshell.
They are, however, all just different varieties of the same plant, like yellow and red tomatoes.
Best Answer
The varieties of the 2 eggplants shown in the pictures are not actually given, but to me the long variety looks like the Japanese/Chinese eggplant and the round variety looks like The Indian eggplant. By the way, there are many different eggplants out there. If you want to have a look, check out this: website.
The Chinese eggplant has a thinner skin, more delicate flavour (I think sweeter) and less bitter seeds, although most modern varieties have bred these out anyway.
The Japanese eggplant is very similar and again has a thin delicate skin and sweet flavour.
The Indian eggplant is similar to the American eggplant with a thicker skin, more seeds and a stronger flavour.