I am wondering what the difference is between House, Electro and perhaps Progressive House in precise musical terms (not in terms of what it feels like/which musicians are examples of the respective genres).
A lot of the defining sound of electro (and related styles) comes from harmonically rich sounds. You'll find distortion and waveforms with many harmonics. In contrast, house tends to be more conservative in that regard.
And that's pretty much it. Electro will also tend to be more "intense", with faster tempos and louder sounds, but I don't think that tendency is a defining pattern since house can also get to those tempos and loudness and electro can be very chill and relaxed.
You could say that, in general, the difference between house and electro is in the timbre of the sounds. This might come as simplistic, but I can't think of another difference at this point of time. There used to be differences in, for example, instruments. Electro used 808s and 909s, but that's not necessarily the case anymore. Sometimes the only difference I find in the genre labeling between house and electro is a sawtooth bassline.
Pattern-wise, progressive house is more trance than house. It uses resources like pads (string-ensemble-like sounds), chord progressions (not very common in house), and loves synthesized melodic lines and time-based effects.
You could say that, in general, the difference between house and progressive house is in the harmonic dynamism (there is more movement, more change in what's going on musically in progressive house) and density (not only more movement, but more things going on, more voices), and effects used (delay and reverb are used not only more, but more prominently in progressive house, for example).
As usual when talking about genres and styles, there are exceptions, blends, mixtures, and labels might (and will) change (sometimes drastically) depending on context and who you ask.
Ok so let me give a try, I might be wrong so don't hesitate to comment below.
Back to the basics, electro comes from electro-funk. Now, there is always a missunderstanding, as Stephen did, between musical genres : in everyone's opinion, house, progressive house, trance, etc... are all under the same barrier : electro or techno. This is wrong. When people don't manage to identify PRECISELY the genre of a song between house, progressive, Drum n' bass or etc, they will name them EDM (Electronic Dance Music) if it's something you can dance on, or "Chill" if it is... well, chill :) .
Now back to your question. House music is known by everyone. It is something that you were used to hear almost exclusively in nightclubs, and it is derivated from disco, that's why the tempo will be +10bpm higher than in dance music in general. Drumly speaking, the specificity of house is a huge, deep, loud kick on every beat, the well known four-to-the-flour! Then it is basically a disco rythm :
Kick ------- Kick ------ Kick ------ Kick -------
------------- S/C --------------------S/C -------- (Snare or Clap)
CHH OHH CHH OHH CHH OHH CHH OHH (Closed High Hat / Open High Hat)
Finally, house music is played with synthesizers most of all, BUT in house music, a lot of samples from funk, soul, gospel or even latina music are used. And THIS is the difference between house music and techno. In techno, they will exclusively use synthesizers.
To end with house music, it is now more and more confused with Electro-house, bigroom or progressive house. This is the more common thing you hear now in nightclubs. Electro-House is way more electronic than house (you don't say?) because it is a merged genre between house and minimal techno. Madeon, Deadmau5 are all a part of this genre. I don't really know the difference between Big Room and Electro-House, but if you want a more precise idea on Big Room, i'll suggest you to listen to Avicii, Swedish House Mafie, Hardwell...
Finally, Progressive House is House + Trance basically.
And to answer to Stephen, to compose an EDM song and respect the conventions is really, really hard. To really well understand this genre would mean a deep historical research, because genres evolved as society evolved.
I am wondering what the difference is between House, Electro and perhaps Progressive House in precise musical terms.
To be facetious, that's like asking about the difference between a straight and winding wooden staircase in precise arboreal terms.
There may be some correlation with the kind of wood you choose for the kind of stresses some architectural design demands, but by the time you arrive at a finished product, the actual building material is rather detached from what characterizes the result.
Bach's "Toccata and Fugue d minor BWV565" has been transplanted into a host of different genres with much of the material from the original score (which would be subject to "description in precise musical terms") remaining identifiable intact.
Electro is a sub-genre of house music. That's all it is. Electro IS house. At least that is what "electro" today is used for. There are several genres called electro, actually.
Electro(funk) is a form of post-disco that developed as a competitor to hiphop. Using optimistic themes to contrast the harsher realities that would later be discussed in hiphop, a sense of romanticism is imbued with notions of adventuring and exploring, as was commonly encountered in science fiction movies and novels. This peaked in the 80s in New York City.
Electro/Elektro is a form of EBM, using uptempo, marching rhythms to contrast the more stomping, sideways dance that characterised EBM. This music developed in the late 80s and saw its biggest rise in the 90s in Germany and Benelux.
Electro(bass) is a derivative of Miami Bass. Artists in the late 1980s wanted to experiment with the electrofunk sound of New York City but using the rhythms of Florida. This also meant using faster and harder rhythms. This music saw its peak in the 1990s.
Nu-electro is again a derivative of Electrofunk but this time it was mostly in Europe in the mid 1990s. Artists wanted the breakbeat rhythm but with the sounds used in trance so this came about. I-F was important as well as Anthony Rother and Carl A. Finlow.
Electroclash is a form of synthpop that drew heavily from I-F's work. Resembling house and garage music, this music came out of New York City but eventually saw Paris, Berlin and Toronto producing many of the big names. Synths sounds were modeled after 80s synths, especially Korg synthesizers (as opposed to Roland synthesizers which dominated the rave scene.) This meant using simple pulse and sawtooth waveforms with various static pulse-width modulations.
Electro(house) is house music that draws its sound palette from electroclash. This music is currently what holds the title of "electro". It started in the early 2000s and the first big hit was Alter Ego's "Rocker". Electro is entirely distinguished by the means of which it is synthesized. It has several affectations that draw nods to chiptunes and other minimal-wave composition methods.
So note the lack of 808s and 909s, as opposed to what others have suggested. Electrohouse typically is a sound that does away with the addiction to Roland synthesisers, instead of prefering the more "retro" sounds of other synthesisers. Some electro tunes do have the thump that resembles the 909, but they are often synthesised differently.