Immoral and unethical are not synonyms, although teasing out the difference between the two can be tricky. It occured to me that a good way to understand that difference would be to come up with areas where they don’t overlap. So …
What’s an example of something that’s immoral, but ethical?
What’s an example of something that’s ethical, but immoral?
I have some ideas of my own, but before I post them I’m curious as to what other people come up with.
By my definitions, immoral means violating my personal code, while unethical means violating my professional or societal code. So…
Billing someone for an hour of work when the job only took 10 minutes would be, to me, immoral but ethical.
A mercy killing could be moral but unethical.
This was done almost word for word about a month ago.
Can we come to a consensus on the differences between ethics and morality?
For starters, you might want to try to read last month’s Can we come to a consensus on the differences between ethics and morality?.
(Or, you may want to avoid it like a bad disease.)
(Your topic is fine. I am only pointing out a possible reason if you do not get a lot of hits on this thread.)
If you hire a fellow to kill your neighbor, because you covet your neighbor’s wife (and her ass,) that’s certainly immoral, on more than one count.
If you skip town so you don’t have to pay the hitman, that’s unethical as well. 
Assuming you really meant:
If we go with the notion that morals come from within (i.e. are personal) whereas ethics come from without (i.e. are societal), then anytime you disagree with society, your ethics and morals are not in alignment.
E.g. If I believe that “all information wants to be free”, copying DVD’s that I rent is moral but unethical. However, if I expect my own intellectual property to be copyprotected, that is ethical, but immoral.
To continue with my above definitions, then anything from being a pot-smoker in today’s Idaho* to being a member of the underground railroad in Antebellum South Carolina* would qualify as such an ethical/moral schizm.