No, duality and symmetry are not the same thing. Although in many contexts "the dual of" is a symmetric relation, this is not invariably the case (e.g. the dual of the dual of a topological vector space need not be the original).
Moreover symmetry is not just about symmetric relations; it has to do mainly with automorphisms of algebraic, geometric or combinatorial structures. Those structure preserving automorphisms (including trivial the identity mapping) form a group, and we'd refer to it as the symmetry group of the structure.
As you note, there are many kinds of symmetry. Some symmetries have order two but many do not. Indeed the group of symmetries may combine elements that have finite order with those having infinite order, elements that have discrete action with some that are continuous mappings. The symmetries of a right circular cylinder, for example, would include discrete actions like reflection in a midplane as well as continuous actions of rotation about the axis.
If you are looking for a fundamental difference, perhaps it should be noted that duality often involves different categories, i.e. the dual may belong to a different category than the original, while symmetry involves not only the same category but actually a mapping of the same object to itself.
Minor nitpick - you should have had $-3 \ln |t|$ instead of $+$.
As to the main question, what's going on here is that $f'(0)$ isn't defined, and the derivative of $f$ approaches $\pm \infty$ at this point. So you'll actually have a vertical asymptote here, and the values of $C$ on either side are allowed to be different, since it does not affect the derivative. Visually, this means that there are two different sides of the function separated by the asymptote, and you can move each side up and down without affecting the derivative. This is why you obtain two different values of $C$.
Best Answer
The difference is entirely a pedagogical one. Both approaches ultimately cover the same calculus.
“Late transcendentals” is the traditional approach to teaching calculus where the treatment of logarithmic and exponential functions is postponed until after integration is introduced. In the traditional method, the natural logarithm is defined by the formula $$ \ln x \;\underset{\scriptscriptstyle\mathrm{def}}{=}\; \int_1^x \frac{1}{t}\,dt, $$ and $e^x$ is then defined to be the inverse function of $\ln x$. Important properties of $\ln x$, such as that $\ln(xy) = \ln x + \ln y$, are proven from the integral definition using $u$-substitution, and the properties of the exponential function follow from these. Arbitrary powers $a^b$ where $a>0$ and $b\in\mathbb{R}$ are then defined by the formula $$ a^b = e^{b \ln a} $$ and it is shown that this agrees with the existing definition when $b$ is rational.
In the “early transcendentals” method, the logarithmic and exponential functions are introduced shortly after the definition of the derivative. Exponentiation where the power is an arbitrary real number is defined by the formula $$ a^b \;=\; \lim_{q\to b}\, a^q $$ where the limit is taken over the rational numbers. (There are some technical issues in proving that this limit exists, which are often ignored.) An argument is then given that there exists a number $e$ for which $$ \frac{d}{dx}\bigl[e^x\bigr]\biggr|_{x=0} \;=\; 1, $$ though again the presentation of this argument is sometimes less than completely rigorous. It follows that the derivative of $e^x$ is $e^x$, and the natural logarithm is defined as the inverse of the exponential function.
The traditional method has the advantage of being cleaner, and the proofs are simple enough that they can be presented to starting calculus students in a mathematically rigorous way. However, it has several distinct disadvantages:
It is not very intuitive, since the natural logarithm and exponential function are essentially summoned out of thin air by what looks like magic. The “early transcendentals” approach, on the other hand, corresponds much more closely with how we actually think of exponentials and logarithms.
Students who take only a single semester of calculus, which includes most biology majors at some colleges in the United States, do not see the exponential function and natural logarithm. This is a serious problem, because these are among the most important functions for applications in biology.
Even students who take two semesters of calculus learn about exponential and logarithmic functions fairly late, which means they don't have time to get used to these functions.
These arguments are generally considered persuasive enough that most universities in the United States have now adopted calculus books that use the “early transcendentals” approach.
Of course, not everyone agrees with this change, and further arguments can be presented on both sides of the debate. Books using the traditional approach continue to be used at some universities and in some calculus classes. In particular, some universities offer an honors calculus sequence designed specifically for math majors and similar students, and there's a relatively strong argument for using the traditional approach in such a course. My impression is also that the “late transcendentals” approach remains quite popular outside the United States, though my evidence for this is purely anecdotal.