見本 is something that has a value being similar to the "real thing" that you can get otherly. Most times in daily life, this is what we expect specimens to be. It thus is not necessary to be an instance of "real things", such as specimens of banknotes or food.
It also has a meaning "good example (for you to imitate)" (= 手本).
標本 is, conversely in some ways, something that has a value being a factual evidence that other things are also like it. For example, a zoological specimen is actually not a "typical example" of that species, but the "definition" itself, where another individual is judged whether being the same species by comparing against it (for this reason, a common species' scientific name is occasionally forced to be changed when they find out the alleged specimen is not identical to what they intended). A boring sample of earth, a moon's stone, and a set of statistical questionnaire data are not "imitation", but actual primary sources for researchers to understand what the whole other part of reality (which is hard to obtain) is like.
We use the loanword サンプル in the same way as English too, but it can often mean 試供品 (freebie or demo).
Specimen for testing is yet another problem: 試料 (or 試験片, 供試体 etc.) in engineering fields and 検体 in medical.