Higher age or older age?

The OP asks which form is correct, not which is most common. So I wanted to give some references from a scholarly text.

A search in all quotations from the electronic version of the Oxford English Dictionary (Second Edition, 1989) of "older/greater/higher age" yields four instances of "greater age", one of "older age", and one of "higher age" (but in the expression "higher age-groups").

• The complete quotation with "higher age" is this:

All show, under natural conditions, the expected ‘population pyramid’, formed by large numbers of young individuals and gradually decreasing numbers of individuals of the higher age-groups.

It seems to me that "higher" refers here to the position of some age-groups within this conceptual pyramid; it isn't qualifying "age", and doesn't mean "older".

• The quotation with "older age" is this, from 1869:

The woman is of an older age than in other described cases [of scleroderma].

This is an example very similar to yours. Personally I consider it acceptable usage since it's quoted by the OED.

• Three quotations with "greater age" use this expression in the sense of "older". I also want to emphasize that they date 1865–1880. However, "greater age" also appears in the definition of "older":

1. In the ordinary senses of old: Of greater age; that has lived or existed longer, of longer existence or standing; more ancient.

Thus it seems to me that "of greater age" is also a completely legitimate expression.

The scarcity and the dates of all these quotations seem to indicate that the two expressions that seem legitimate are nevertheless rarely used.

That said, the final judgement is yours and depends on your personal views about language and standards.

Here's a snapshot of the search results

"Of older age" is chiefly an Americanism/ of newer usage. (Opinion:) I would think it's obviously incorrect, though, much like "at *faster speed" and such.

Otherwise, it has mostly (and appropriately) been "of greater age," (opinion:) which I would recommend singularly — sadly, it's now losing to the older alternative everywhere.

Though "of higher age" is in use, especially in certain contexts, it's not the preferred phrase.

I don't think it is common to say either higher age or older age. People normally say (in your case):

older/younger patients

Or patients of the same age, just like what Max said in the comment above.

But I also think it is grammatically correct to use patients of higher age or of older age (it's just not preferable to use). Apparently though, based on Google Trends, the phrase older age is more popular than higher age.

Older age is quite popular in the UK, US, Australia, India, Canada and South Africa. Whilst higher age is merely used in the UK, US and India.

Therefore, my conclusion is:

older patients is better to use than patients of older age than patients of higher age.