You now find a lot of people using admixture as a synonym of mixture; it is so widespread that this cannot be regarded as an error.
But the classic use of admixture is narrower: the prefix ad- (Latin = “to, toward”) implies that a new component or ingredient is brought “to” an original substance and mixed “into” it. We might for instance speak of the classic Manhattan cocktail as
rye whiskey with an admixture of vermouth and bitters.
In the phrases "any relation" and "no relation", we always use the word "relation".
Is he any relation of yours?
John Smith was friends with Brian Smith (no relation).
(We would not say "any relative". It would, however, be correct to say "Is he a relative of yours?" or "not a relative" - though in the latter case, "no relation" is more idiomatic in most contexts.)
In a few expressions, such as "cared for by relatives", it is virtually always "relatives".
Otherwise, the two are synonyms, but "relative" is the more common term.
In "relative by marriage", "close relative", "distant relative", "my relatives", etc, the term "relative" is significantly more common, but "relation" is equally correct. It is also very much more common to refer to "visiting relatives" than "visiting relations".
Of course, there are other (non-family-related) definitions of both words, so they differ in that respect, too.
Best Answer
I would say they are pretty close, but a seaside resort might encompass more than a beach. So, you might have taken a seaside vacation, and never felt the beach sand between your toes. Instead, you might have participated in activities such as golf:
Your second sentence, therefore, is more specific, and makes it clear that you weren’t merely at the seaside, but on the beach: