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Rabbi A says strawberries are ha'adama; Rabbi B says that strawberries are Ha'etz.
The halachic texts are replete with such differences in "opinion". What's the underlying reason that different rabbis can arrive at a different conclusion for what to do in scenario X?
Is it because each Rabbi uses his own understanding mixed with the rules of deriving halachah which allows for error? If so how can we follow anything they say?
Besides for the difference in applying identical methodologies detailed by @y.lub there are also differing methodologies used. For example, Rambam relied heavily on the Talmud Yerushalmi while most other Rishonim ruled almost exclusively from the Talmud Bavli. Furthermore, some poskim rely more on their own conclusion based on Gemara, others (admittedly rare) rely primarily on responsa of Geonim, others rely primarily on rulings of Rishonim, while others tend to follow the Shulchan Aruch / Rama and / or commentaries thereon. There are also differing approaches to minhag. Some poskim (e.g. Aruch Hashulchan nad R. Mosshe Feinstein) utilized minhag as an authority in and of itself to a much higher degree, while others (e.g. Mishna Brura) do so to a much lesser degree.
See "The Making of a Halachic Decision" by R. Moshe Walter for citations and analyses of various views.
Other answers give technical reasons why two people can come to different decisions regarding the same issue. I once read an excellent "common sense" explanation which helped me.
A school leadership team discusses whether or not to expel a student from school for misbehavior. One group argues that learning is negatively impacted by the student's behavior, that expelling him is required to preserve the other students and that nothing else they tried has worked.
The other group argues that one should give the kid another chance, that he will never find another school and that expelling him is the best way to ruin his future.
Both are valid lines of thought. Both represent different philosophies about life and education. Both are "right" in many ways and "wrong" in some ways.
At the end the school principal will take a decision. But that decision doesn't mean the other line of reasoning is faulty. However in the real world one needs to move forward and he can only pick one path.
Such is different paths in halacha. They might represent different philosophies, or apply to different circumstances, or different times. Picking one path doesn't invalidate the other one. Elu v'elu divrei elokim chayim.
+1 excellent question. This topic is one of the reasons I enjoy learning so much.
There are a number of variables that can go into a difference in opinions in Halacha. Starting with the earlier sources, variant readings in the Gemara and different interpretations of what exactly is going on in the Gemara and how to resolve contradictions will affect outcomes. For example, one Rishon might answer a contradiction by making a certain אוקימתא or establishing that the Halacha is so only in a specific case, while other rishonim will reject a Gemara from being definitive in light of another and not be compelled to resolve the contradiction to make both gemaras fit with Halacha.
Another possible reason for a dispute can be found in logic and when to apply certain concepts or whether to derive one Halacha from another (מדמה מילתא למילתא). One authority may bring a proof to a certain דין and another may refute that proof.
THis question is so important I think it should be marked sticky or FAQ or else - this is the base for the rest of the site!
THe previous answers address the question of "HOW do they arrive at different conclusions" not "WHY", I'll try to address the question of WHY.
The Torah, which is the blueprint of this world, is contradictory and hence the world is contradictory as well - it is by design! The first Rashi on the first word of the Torah says it explicitly - "בשביל תורה שנקראת ראשית ובשביל ישראל שנקראו ראשית". Sometimes the Torah will take precedence, sometimes Yisroel.
Even if we imagine for a moment that there's a "true solution" for any Sugyah - since the תנורו של עכנאי incident with R' Eliezer and the Sages, where they ruled explicitly against the heavenly truth, haKABA"H does not care about the truth anymore, as long as they arrived at their conclusion sincerely and wholeheartedly.
Therefore it is true to say that the Halakhah is not about the truth, it is all about engaging in Torah study - that's all that matters.
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It is the reason, the interpreters / Poskim are not limited to any methodology, and not limited to address all the opinions known to them (see haMechaber choosing only 3, for example). In other words "they don't really care", as the process is one that matters, not the results.
THere's a strange and unexplained tradition of finding a Tirutz instead of an explanation. Although you can never tell the difference, a Tirutz is not meant to be a full answer, just a possible special case scenario. Those Tirutzim may vary greatly and contradict each other.
Those are the reasons I can think of.
