Membrane
Trilaminate and Membrane are both ways of describing the same type of suit. Two layers of material, usually nylon or polyester are bonded together with an inner waterproof layer, usually butyl.
They provide little thermal insulation but are light and flexible to wear whilst being extremely robust and long lasting. An undersuit is required even for short shallow dives.
They don't stretch, so a good fit is important for safety and comfort. Too tight and they will restrict movement making access to valves difficult and making it difficult to climb back on to a boat.
Too loose and the large quantity of trapped air will cause buoyancy problems. At worst, if the legs are too long the boots may slip off and cause the wearer to become inverted.
Good quality membrane dry suits are reassuringly tough and reliable, comfortable and can last a very long time.
They can be made from breathable material for surface use such as sailing and canoeing.
Neoprene
Made from closed-cell synthetic foam rubber. They stretch and provide some insulation.
Thick neoprene suits 6 – 7mm are a good budget buy as they are tough, reliable and easy to repair with Aquaseal® or Black Witch. You don't need an expensive undersuit, just a base layer or jogging suit.
The downside is, they are restrictive to wear and they loose insulation as you descend and the neoprene is crushed. Buoyancy can also be tricky as they spring back to shape as you approach the surface.
They are long lasting, but gradually loose insulation and buoyancy as the neoprene ages.
Not ideal for deeper diving but they should be cheap to buy and are usually a better bet than a cheap or second-hand membrane suit.
Thin neoprene suits 2 – 3mm are stretchy, comfortable and slim fitting so they look good and don't have inherent buoyancy problems. They are not very warm, so you need a good undersuit. The material is not very tough, so the life expectancy is limited. Seam sealing is more difficult than with thicker suits and they are tricky to repair, so you definitely do not want to buy a cheap one from a company without a lot of experience.
Genuine crushed neoprene suits are tougher and longer lasting, but don't stretch much. They are difficult to manufacture and repair and are always very expensive.
In between is a huge range of 4 - 5mm suits. The best will be made from compressed or high density neoprene. This retains some insulation at depth and doesn't cause any buoyancy issues. They have a useful amount of stretchiness but are not as tight fitting a thinner suits.
There is a huge amount of advertising hype applied to neoprene, most of which is rubbish.
Cheap suits will soon leak.
Good ones are comfortable, reliable and long lasting, but not cheap and don't last as long as good membrane suits.