Strength & Hardness

Generally speaking, the strength of a wheat is dictated by the quality and quantity of the protein.

Strong flour produces a good elastic gluten at the dough stage and has good gas retention properties which is favoured for breadmaking as it produces loaves with good crumb structure and volume.
Weaker flour produces extensible (ie non-elastic) doughs and is preferred for biscuit and cakes.
The difference between hard and soft wheat relates to the texture of the endosperm. Hard wheats, when milled, produce gritty regular sized granules which sieve well and flow easily and starch damage is easy to achieve due to the brittle nature of the starch granules. A cleaner separation of bran and endosperm makes milling easier and produces greater extraction rates.
Soft wheats produce irregular granules which are soft in nature and tend to be squashed rather than crushed in the milling process; These characteristics lead to poor handling in the milling process, more difficult separation of bran and endosperm and lower starch damage capability.
Starch damage caused as a result of the milling process is the controlling factor in the amount of water that can be absorbed in the dough making process, and dictates the yields achieved in the bakery.
Wheat quality is therefore defined differently depending on whether you are a farmer, miller or baker. Growers are looking for high yielding wheat, millers look for hard, vitreous grain to aid high extraction, and the baker is looking for low enzyme activity flour with high starch damage (to improve absorbtion). The target for breeders is to develop wheat varieties that meet all these requirements.
So to summarise - the strength of a wheat dictates its' baking potential, and the hardness dictates its' milling potential. Both these characteristics are independently identifiable in the breeding process. The breeders therefore can develop strong/soft or weak/hard or strong/hard to meet the needs of millers and bakers at the same time.