Differences in engines

The 1.4>1.6

Both are TU block (cast from the same mould, however there are differences) With the 1.4 being a TU3 and 1.6 TU5
The major difference is an increase in stroke and bore. Early 1.4s can be found carb'd but they are few and far between.
The pre 1997 1.4's are all SPI (single point injection) ie, single injector units. All 1.6's and all post 1997 1.4's are MPI (multi point injection) ie 4 injectors. Basically means a more economical and more responsive engine.
The TU block was never really meant to be made into a 1.6. Most pre 1990 Peugeot 1.6 were XU blocks. It was quite a good engine, the finest of all TU engines came in the pre 1997 106 Rallye. One of the best engines Peugeot have ever produced and very similar to the legendary 1.3 205 Rallye which was never seen in RHD format.

1.6>1.8

Totally different, different block types altogether. The early 1.8 8v's were awful wheezy units, the 1.6 wasnt far behind in terms of performance.

1.8>2.0

2.0 has a longer stroke and a slightly bigger bore than the 8v, apart from that, the engines are almost identical. Same story with the 16v, the engines are far better. Breath more easily being 16v and rev higher. The change to 16v happened in early 1997. The 1.8 showed the most benefits as it was a vast improvment over the older 8v model in both performance and economy.

Dturbo>HDi

Very different. The Dturbo unit is a XUD, so its a XU block. Its an old ricketty engine but will last forever. The key to these engines is finding out which fuel pump you have (Bosch or Lucas) Then you can adjust the fuel pressure to give significant gains in power and torque. The word is the Peugeot had to de-tune the diesel turbo XUD units because they were almost on par with 2.0 petrol performance wise and cost less to buy! Hence the easily tuned 92bhp units.
The HDi is/was the first of the DW10 engines to make and appearance. It was the only EW engine put in the 306. The XU engine was abolished all through the range in 2001 with the end of the 306. The EW engines are much more refined, more economical, more powerful and more environmently friendly than its older XU brother.
The TU block is still going strong and can be found in 107's, 206's and 307's.

S16>GTi-6

The most commonly asked question.
The S16 XU10J4 engine is loosley based in the older 405 T16 engine (XU10J4TEL). Its the very same engine as seen in the post 1991 405 Mi16. It has an ACAV breathing system which is basically a poor mans VTEC. Larger butterflys are opened on the inlet to allow more air in once the engine reaches a certain rpm (approx 4500) However the system is prone to failiure as the diaphragms which control the opening of the butterflies is very weak and often fails. So performance would be limited.
The S16 has a very good bottom end, it retains the knife edged crank shaft from the T16 with balanced rods makes it a fine thing indeed. The exhaust manifold is the weak point of the engine.
The GTi6 uses the same block with some major changes. It doesnt have the superb bottom end of the S16 but.. Its gains come in the head. Its totally different, lightened valves, higher lift cams, vernier pulleys, accousticlly harmonic inlet with 4 branch tubular exhaust manifold, twin injectors, piston cooling by oil jets.... I could go on...
Basically the S16 was a great car in its day, it was surpassed by the GTi6. Evolution.