Other than the differences in spelling, different nuances of sense are expressed to those aware of the historical application of the terms, or simply those sensitive to the different uses of the words. These nuances, connotations that go along with the words, are meaningful, and to some people, very meaningful.
First, observe that 'lead' in this sense may indicate the dog is in the lead, not that the holder of the lead is in front of the dog. In the case of coursing hounds, for example, three to a leash perhaps, chasing a hunted animal, the hounds are leading the hunter toward the animal.
Observe these differences. Under the headword lead, n. in the OED, we find this
- concr. Something that leads.
d. A leash or string for leading a dog.
["lead, n.2". OED Online. September 2015. Oxford University Press. http://www.oed.com/view/Entry/106585?rskey=M5Br57&result=3&isAdvanced=false (accessed September 13, 2015).]
With the definition are two historical quotes exampling that sense of 'lead' in 1893 and 1898.
Under the headword leash, n., however, we find this wealth of meaning:
a. The thong or line in which hounds or coursing-dogs are held. ....
- A set of three; originally in Hunting, used of hounds, hawks, foxes, hares, deer, etc.; hence gen.
- Hawking. The thong or string which is passed through the varvels of the jesses to secure the hawk.
- fig. (with allusion to senses 1, 3); esp. in phrases, to hold or have in leash, to have control over, keep in bondage.
["leash, n.". OED Online. September 2015. Oxford University Press. http://www.oed.com/view/Entry/106745?rskey=VIzxKU&result=1&isAdvanced=false (accessed September 13, 2015).]
With the definitions are examples of uses dating back to before 1300 for 1a, around 1330 for sense 2, 1497 for sense 3, and around 1430 for sense 4.
As you can see from the different senses, the use of 'leash' may carry along with it many connotations which the use of 'lead' does not. While it is probable that many, if not most, people hearing the terms will be unaware of the differences, many people will be aware of them. Those people who are aware of the differences, and especially those who use 'leash' in the more specialized senses, may find the general use of 'lead' to mean 'leash' in its more specialized senses distasteful or simply imprecise.