What was the reason for the phasing out of cutting weapons like long swords for thrusting weapons like small-swords and rapiers? It seems to me that I would rather have a heavier weapon that I could someone more easily with than a small blade that I could miss with more easily. The 'pistol vs French grip for fencing' thread, got me curious about this when someone mentioned that cutting weapons went out of style in favor of thrusting weapons.
They didn't. The cutlass continued in naval use. The British cavalry sword didn't commit to a pure thrusting weapon until 1908.
In the case of British cavalry swords the switch was due to a long running argument. A thrusting sword would give the best chance of a quick kill during the initial impact of the charge but was rather limited outside those conditions. During a general melee a slashing sword would be better. For most of the 19th century compromise weapons were used that could both stab and thrust to an extent. However such compromises had their limits and in 1908 a pure thrusting weapon was chosen for maxim initial impact. Just in time for sword armed Calvary to lose much of their remaining effectiveness.
Long swords are themselves a compromise weapon. Pure cutting weapons have wider blades and are often curved. Rapiers are around 50% lighter while being much the same length which lets you move faster and is just that much less to carry around. They are also one handed weapons which improves reach which is really rather useful if you aren't wearing armour. You've also got the shift from swords being a general battlefield weapon to having a more limited role with the focus in duelling.
A largely two handed sword like the longsword is going to perform poorly against someone who knows what they are doing with the rapier. Not that everyone was happy with the rapier. George Silver writing in his Paradoxes of Defence in 1599 argued against it in and favour of the backsword (which had some limited cutting ability although Silver favoured thrusting).
By the time the small sword appears on the scene swords had effectively become an element of masculine jewellery (there were entire industries that produced decorated sword hilts). While an effective weapon in the hands of someone who knew what they were doing there was certainly an element of trying to chose a weapon that was practical to carry around all day.
As the other user said they did not. They just diverged from "duelling weapons" in the face of the smallsword and the later rapiers. Cutting weapons have remained a preference for war. Contemporary with the rapier (which itself has surprising cutting power and was used on the battlefield) are various backswords, baskethilt swords, schiavonas, etc. (if you just google each name you will easily find examples). Sometime in the 17-th century sabers start to enter the field (well they have been around in Eastern Europe since 8-9-th century), only to become a very widespread weapon and remain in use ... even today.
Now, this is all military. I am not exactly sure about why civilian weapons became more thrust oriented, as the cut vs. thrust has been a century old debate. Yet you see the roots of these changes as far back as I.33 - the oldest fencing manual we have. They are in full force, and you can literally see the change as you read through the Bolognese masters, starting with more cut-oriented swords with Antonio Manciolino (his book is from 1523), and getting pretty stabby two-three generations later with Giovany Dall'Agocchie (1572).
Some of the old dead fencing masters argue that the thrust is faster and safer, since the blade travels only in straight line, and it is easier to cover yourself. On the other hand a cut has more stopping power, when delivered correctly. At least in rapier fencing it is still present in quite late treatises throughout the 17th century, although used to a lesser degree. I think it is even there in smallsword treatises, though it correspondingly shares an even smaller proportion of techniques