This is tricky one, I'll try and explain my perspective as a native British English speaker. I have a biology background as opposed to physics, so that might come into play as well here.
Structure: Arrangement of and relations between the parts or elements of something complex or a piece of construction.
Morphology: A particular form, shape, or structure or the study of something's form of shape.
As you can see, they both have one usage where they are near enough synonymous, and another where they are not. In most situations they can be used interchangeably (if you mean the common usage) and you will likely be understood but there are some contexts when the other word would be more correct. I find those cases tend to be when discussing 'natural/developed/formed' vs 'unnatural/constructed' forms/structures. Structure is more about describing the network and geometry of components in an abiotic way.
The animal's morphology or The animal's structure.
In the first sentence you are clearly talking about the macro structure of the animal. You are talking about it's skeleton and muscle arrangement, it's shape and size e.g. it has a tail, it has large eyes, it has pentadactyl limbs.
In the second sentence, if nothing else, as structure has the 'construction / building' noun usage this sentence is ambiguous. You could be refering to something the animal built. Even in context, this wouldn't work as well as 'morphology'. I'd only use this if we were talking about an animatronic animal. Interestingly, if you were talking about the animal's kidney, you could use wither structure or the morphology.
The building's morphology. or The building's structure.
Here, I find 'structure' a much better word.
The earth's internal structure or The earth's internal morphology
Here, you could use either.