The dictionary definitions for 整个 and 全 both use "whole" and "entire". However, 全 additionally has definitions of "all" and "complete".
I have a homework assignment that asks me to distinguish their usage in various sentences, but I don't recall there being any significant difference between the two.
My intuition tells me that 整个 would be used for something with obviously discrete parts, such as 整个美国人. I would use 全 for other things like 全身体.
Is this the right intuition?
Thinking of the literal translation helps.
整 is 'entire, whole'. 个 is a widely applicable singular measure word, hence 'unit'. So 整个 is, literally, 'the whole unit (of), the entirety (of)', as opposed to 'a proportion (of)'.
我整个身体都不舒服。My whole body feels sick.
However, I don't think 'whether the object is having parts or not' matters for applying 整个 (on the contrary, it matters for 全). In the following examples, 'the ground' and 'the afternoon' are not 'obviously decompose-able':
白雪覆盖了整个大地。The entire ground is covered by white snow.
他一整个下午都在看书。He was reading books for the entire afternoon.
全 means 'all, every'. It is applied to a collection, or a unit that has 'obviously discrete parts'.
In this sense, as the other answers already mentioned, 整个身体 means 'the entire body' and 全身 means 'all parts of the body'.
我全身都不舒服。All parts of my body feel sick.
(As for why 全身 is used instead of 全身体, it's probably because 全, 身 and 体 are all bound morphemes, and people tend to construct words with the least number of bound morphemes necessary.)
As you may notice in this example, although 全 is applied to a collection, the word after 全 is not always a 'group noun'. Actually in most cases, it is only in the semantic but not the syntax/word formation. For example:
全球金融危机 The global financial crisis
他全天都在工作。He is working all day long.
In these phrases, the knowledge (in the extent of Chinese language, not philosophy) that 'the globe' and 'the day' can be further broken down is implicit. On the other hand 'The afternoon' is considered atomic, so you can only say 整个下午 but not 全下午.
I suppose you are right that obviously discrete components are involved with 整个. For instance, 整个部门 is the department as a whole, i.e. all the employees in a department. 个 does the unifying job because it implies what follows is a unit.
With 全, however, the emphasis is shifted slightly onto the collection, a plural, e.g. 全国 seems to me to be more 'all across the nation' rather than 'every state in the nation'. Possible exceptions are smoothed out.
By the way, note that 全身不舒服 is discomfort throughout the entire body, all parts of the body, and one does not use 全身体 ... it just feels wrong, I can't explain it right now.
I'd be glad to learn more about this, it's the blind spot a kind of native speaker faces.
我整个身体都不舒服and我全身都不舒服are both correct. 整个 implies you treat something as one, e.g. 整个人, 整个国家, 整个民族; 全 implies "all of discrete parts". So全身means all parts of the body but整个身体means the whole body. And you can't say 整个美国人, but 全部美国人 or 全体美国人.