That's actually quite common. For a lot of drugs, the place that manufactures brand-name drugs will also manufacture (some of) the generic versions available. (This may or may not be operated directly by the pharmaceutical company itself).
For really common medication, there will be multiple plants that manufacture the drug. But for less-common drugs[0], it's possible that there is only one place that manufactures them.
> So it's hard to claim that there's even a minor difference, unless the company specifically manufactured them differently
Quality control, for one. Generics are allowed to have a rather wide margin of error on the amount of the active ingredient they contain, for example. Brand-name drugs tend to keep their range much tighter, because they want to protect their brand.
EDIT: To clarify the statement "wide range of error", there are a couple of different ways of looking at it. Think of mean and variance: a drug is allowed to have varying amounts of drug from pill to pill, and the distributions of the active ingredient can have different first (mean), second (variance), and third (skewness) moments in the generic and brand-name versions[1].
But on top of that, there's the question of measuring the active ingredient by mass in the pill, vs. measuring the bioavailability of the drug (ie, how much actually is available to the body to use). The latter is the thing we actually care about. The problem is that defining that is a tough problem, because it differs from person to person, and some inactive compounds affect absorption in some people but not others. Short of literally manufacturing the generic and brand versions simultaneously and randomly assigning them as they come off the belt[2], there is no way to guarantee that bioavailability will be identical between the two drugs for a specific patient. If they have different inactive compounds, or if any part of the manufacturing process differs in any way, they will be different for some subset of patients, and there's no amount of testing that can change that basic fact.
After a drug is approved and becomes generic, the brand name and generic versions are subject to the same rules[3]. The generic drugs aren't "different" from the brand name in this respect; it's just that the brand name is a known entity, whereas it's hard to request a specific version of a generic compound. So if you know that a brand-name version works, it can be preferable to stick with that, because you can always request a prescription to be filled by the brand-name ("dispense as written").
[0] and this is a very long tail, so it's quite likely that a drug you take or have taken falls into this category
[1] To be pedantic, we know from continuity theory that they have to be different, so the question is how different do we allow those to be?
[2] Which, to be fair, is not too far from how some generic drugs are made, but this is completely opaque to the end-consumer and subject to change without notice.
[3] http://www.fda.gov/downloads/AboutFDA/Transparency/Basics/UC...