Eastern concept is that MOST of your Karma effect happens in your next life or contributes to the type of life you will have next. Western concept is that it happens in this life.

If by classical cause and effect, you're referring to Newtonian physics, in simple terms, one major difference would be the description of an ethical dimension to cause and effect. And the correct term is Karma Vipaka, meaning action and it's results (vipaka means fruit or result) as Karma by itself simply means action. Hinduism uses the term Karma more to refer to duty and obligation, hence the distinction of using Karma Vipaka. The difference then can be said to be that while cause and effect in Newtonian physics refers to properties of physical objects, is mechanistic and doesn't account for actions at a distance, Buddhism uses Karma Vipaka to indicate cause and effect in a much less mechanistic sense, specifically related to volitional action. A better understanding of how at least the Buddhist use of the term karma vipaka equates to modern science is to understand quantum physics view of causality, which deals much more in terms of probability than Newtonian physics does, is not mechanistic, involves the effects of consciousness and observation, and again has room for action at a distance. If you have specific interest in this area, Bell's Theorem is essential reading, especially with regards to locality and probability. I can also recommend the work of Ehei Dogen, the thirteenth century Zen master who's master work (Uji - On Being and Time, in the Shobogenzo) deals specifically with the relativity of space time, and some of the other fascicles in the Shobogenzo that are on the non local nature of Karma Vipaka. And, to hopefully simplify this issue, and to show where the dividing line is at least between the Buddhist teaching of Karma and scientific understanding of cause and effect, this quote may be of use: "The Pali word kamma or the Sanskrit word karma (from the root kr to do) literally means ‘action’, ‘doing’. But in the Buddhist theory of karma it has a specific meaning: it means only ‘volitional action’ not all action. In Buddhist terminology karma never means its effect; its effect is known as the ‘fruit’ or the ‘result’ of karma. The theory of karma should not be confused with so-called ‘moral justice’ or ‘reward and punishment’. The idea of moral justice arises out of the conception of a supreme being, a God, who sits in judgement, who is a law-giver and who decides what is right and wrong. The theory of karma is the theory of cause and effect, of action and reaction; it is a natural law, which has nothing to do with the idea of justice or reward and punishment. Every volitional action produces its effects or results. If a good action produces good effects, it is not justice, or reward, meted out by anybody or any power sitting in judgement of your action, but this is in virtue of its own nature, its own law. This is not difficult to understand. But what is difficult is that, according to karma theory, the effects of a volitional action may continue to manifest themselves even in a life after death..." - Ven Walpola Rahula, What the Buddha Taught