It is worth pointing out what exactly enantiomers are: exact mirror images of each other. Most of what enantiomers do and don't do can be understood by remembering that basic definition and applying it logically.
First, let's compare a chiral compound to an achiral one. Imagine an achiral molecule in front of a (molecular-sized) mirror. The orientation does not matter; the mirror will show a mirror image. Take a screenshot of the picture in the mirror. By definition, it will be possible to take that achiral molecule and rotate it so that it looks exactly like the screenshot you just took, except that you aren't using the mirror.
Whichever macroscopic property of the pure material you decide to measure, the resulting value is typically one of two things: either a simple number with a unit (i.e. it is a non-directional property; e.g. melting point) or a directional vector (e.g. rotation of polarised light. Each individual molecule will contribute identically to the simple-number properties; whereas the directional properties will typically be averaged out as there will always be a molecule oriented exactly opposite in a large enough sample.
Enantiomers on the other hand cannot be mapped onto their mirror image. Thus, it will be impossible to find a molecule oriented in exactly the opposite way in a pure sample of a single enatiomer. However, you are able to produce an equally pure sample of the opposite enantiomer, i.e. the exact mirror image. Now, the simple-number type of macroscopic properties will remain unaffected by which enantiomer you are measuring but those properties that are vector-based will be opposite, as they cannot be internally compensated.
Therefore it follows that unlike your first sentence enantiomers do not have similar properties but identical ones save those where direction actually matters. The most common example of an experiment where direction matters is the rotation of polarised light.
However, this difference is a consequence of the molecular structure; not the other way around. It is worth remembering that.
Now we move on to the interaction between a chiral compound and another compound, be it achiral or chiral. You can do some macroscopic experiments to investigate this, if the thought experiment isn't conclusive enough. For example, take a (small!) figure or sculpure or whatever of a human being or animal whose arms or legs are doing different things. This could be your chiral molecule. As an achiral object, you could take e.g. a mug. Play around with the two to see how they interact. You will be able to get 'better fits' one way but not the other way. For example, if your figure has one arm bent at an angle while the other is straight, this angled arm can probably easily lock around the mug's handle while they are both standing on the table; not, however, using the other arm. This is akin to a chiral molecule reacting with an achiral one in a stereospecific manner.
To understand how two chiral molecules interact, use two such asymmetric figures and see what you can do. One way will 'fit' notably 'better' than the other way. Finally, image one of the figures being a mirror image of itself. Now, the interaction will be markedly different and they may not even fit together at all. (Of course, it would be ideal if you had some kind of mirror images of figures but I'm not sure if that's all that common. Some kind of irl 3D tetris blocks perhaps?)
This is the gist behind the image in Tiberius' nice and succinct answer. The ramblings herein only serve to further illustrate the point and to allow a thought path to be re-understood.
Finally, I will briefly address that Alchemista's comment can also be understood in this context but I will argue that Alchemista's examples are not physical properties of the pure compound and thus not really applicable to the first part of this answer.