We study enantiomers in detail. But what makes them more important to study than, say, diastereomers? What I'm trying to ask is, enantiomers are simply a type of stereoisomers that can rotate plane polarized light in opposite directions by the same amount. What makes them more important than other stereoisomers?
The reason (strict mirror image) enantiomers have more attention paid to them than diasteriomers (at least in Organic Chemistry classes) is because they share the same reactivity toward achiral reagents.
To achiral compounds, both of the enantiomers look the same. There is no way for the achiral substance to determine the difference between the two. If you take a reaction with an enantiomer and flip it through a mirror, there's no chemical difference between the two.
In contrast, there are a number of effects which cause diastereomers to behave differently. (Basically, the chiral centers in one part of the molecule interact with the chiral centers at the other part, and because those pairs are different in two different diastereomers, the reactivity is different.) The upshot is that two diastereomers have as much difference in behavior as compounds which don't share molecular formula. (e.g. the difference in behavior of an ethyl versus methyl variant.)
This has very real implications for practicing organic chemists. If you start with achiral starting material and produce a chiral compound, there is no way (with strictly achiral materials) to produce anything but a racemic mixture of two enantiomers. You can futz with conditions to get specific diasereomers, but not specific enantiomers. In order to get an "enantiomeric excess" of one, you either need to use chiral reactants or a chiral catalyst.
Likewise with separation. If all you have is an achiral chromatography column (e.g. silica) and simple (achiral) solvents, you're not going to be able to separate two enantiomers. They're always going to migrate the same. Again, futzing with conditions can get an achiral chromatography column to resolve diasteromers, but not enantiomers. To resolve enantiomers, you need a chiral column (or less likely, a chiral solvent).
Combined with the importance of particular stereoisomers in biological systems, that's why the emphasis on enantiomers in Organic Chemistry classes. Historically, it was a big sticking point in the development of organic synthesis to a) figure out that enantiomers were a thing, and b) work out how to deal with enantiomers. Even today there's tons of research into chiral control. However, modern practicing organic chemists tend to work on larger and more complex systems, with many stereocenters so the enantiomer/diasteromer divide isn't as important, as often they're working with stereocenters "in isolation".
Enantiomers have different physical arrangement of atoms. This difference plays a crucial role in biology.
But, why do enantiomers have different biological activities?
Recognition of chiral drugs by specific drug receptors is explained by a three-point interaction of the drug with the receptor site, as proposed by Easson and Stedman. The difference between the interaction of the two enantiomers of a chiral drug with its receptor is illustrated below.
The main reason enantiomers are emphasized more comes down to the interesting properties they have, particularly in biological systems.
A pair of enantiomers exhibit identical physical properties (e.g. melting point, dipole moment, etc) except for their optical activity (as you said, they rotate plane-polarized light with the same magnitude, but in the opposite directions). Chemically, however, a pair of enantiomers can behave very differently. This is particularly seen in biological systems, which are often adapted to work properly with only a particular enantiomer of a pair. For example, almost all naturally occurring sugars are the D-enantiomer and almost all naturally occurring amino acids are the L-enantiomer.
This has major implications in medicine; often times only one of the enantiomers is effective and the other can be inert or even harmful. The comments on your question mention thalidomide, a drug that was sold as a sedative and a treatment for morning sickness until it was realized that while one of the enantiomers had the desired effect, the other caused birth defects. Making enantiopure mixtures is necessary to ensure you are truly generating the effect you want (in the case of thalidomide, this still fails because the compound naturally racemizes).
For diastereomers, there is a greater expectation that they would behave differently due to their different physical properties. Hence, a lack of emphasis on diastereomers (which I'm not certain there is a lack of emphasis, at least from the organic chem course I took) may be due to the fact that the relationship between diastereomers is less direct than that between enantiomers and so it is tougher to draw conclusions about what properties they will share.