You are correct to observe that with many conditionals, such as the one you quote, "if P then Q" is contrary to "if P then not Q". It is true of conditionals that express causal relationships, and those that describe dispositional properties, and it is true of counterfactual conditionals, among others. Fundamentally, this is because a conditional is not so much a sentential connective but a device for introducing a suppositional context. "if P then Q" can be glossed as, "Suppose P... In that case, Q". If you assert, "if P then Q" and I agree to accept your supposition P and disagree with you about whether Q is true under that supposition, then our beliefs are contrary.
There are conditional logics built around the principle that "if P then Q" is contrary to "if P then not Q", such as connexive logic. It represents a strand of thought in the history of logic running from Aristotle through Boethius to Abelard. It is also worth noting that conditional probabilities have this connexive property: if P(B|A) is high then P(¬B|A) is not high.
However, this does not hold true of all uses of conditionals. Sometimes we wish to advance a supposition P that may or may not actually be true, because we wish to consider what would follow if it were. Reasoning by reductio follows this pattern. If we suppose P and it leads to a contradiction then we may conclude that P is false. In such cases, "if P then Q" and "if P then not Q" are both correct and they conjointly imply not P. Conditionals like this are very useful in mathematics and formal logic, because often we want to start from a bunch of hypothetical assumptions and 'solve' for P.
The simplest conditional that has the latter property is material implication (→). From P → Q and P → ¬Q we can prove ¬P. Negating a material implication gives P ^ ¬Q, so we might say that the semantics of material implication is that P, ¬Q are not both true. By contrast, the negation of a connexive "if P then Q" is "if P then ¬Q" which suggests a stronger relation of incompatibility between P and not Q.
When you begin to study logic, material implication is the first conditional you are introduced to, because it is the simplest and the only one that is a truth function. It also has a special place in classical logic, because by the deduction theorem it is the sentential connective that corresponds to the classical relation of entailment. But it is important to remember that it is not the only conditional. Indeed, I would go so far as to say that the great majority of uses of 'if' in English cannot be correctly translated by material implication.