Wikipedia describes many variants of quantum field theory:
Are these approaches to the same thing or actually different?
They're variants, different kinds of quantum field theory, but they're not mutually exclusive. The different adjectives you mention separate quantum field theory to "pieces" in different ways. The different sorts of variants you mention are being used and studied by different people, the classification has different purposes, the degree of usefulness and validity is different for the different adjectives, and so on.
Conformal quantum field theory is a special subset of quantum field theories that differ by dynamics (the equations that govern the evolution in time), namely by the laws' respect for the conformal symmetry (essentially scaling: only the angles and/or length ratios, and not the absolute length of things, can be directly measured). Conformal field theories have local degrees of freedom and the forces are always long-range forces, which never decrease at infinity faster than a power law. They're omnipresent in both classification of quantum field theories - almost every quantum field theory becomes scale-invariant at long distances - and in the structure of string theory - conformal field theories control the behavior of the world sheets of strings (here, the CFT is meant to contain two-dimensional gravity but the latter carries no local degrees of freedom so it doesn't locally affect the dynamics) as well as boundary physics in the holographic AdS/CFT correspondence (here, CFTs on a boundary of an anti de Sitter spacetime are physically equivalent to a gravitational QFT/string theory defined in the bulk of the anti de Sitter space). Conformal field theories are the most important class among those you mentioned for the practicing physicists who ultimately want to talk about the empirical data but these theories are still very special; generic field theories they study (e.g. the Standard Model) aren't conformal.
Topological quantum field theory is one that contains no excitations that may propagate "in the bulk" of the spacetime so it is not appropriate to describe any waves we know in the real world. The characteristic quantity describing a spacetime configuration - the action - remains unchanged under any continuous changes of the fields and shapes. So only the qualitative, topological differences between the configurations matter. Topological quantum field theory (like Chern-Simons theory) is studied by the very mathematically oriented people and it's useful to classify knots in knot theory and other "combinatorial" things. They're the main reason behind Edward Witten's Fields medal etc.
Axiomatic or algebraic (and mostly also "constructive") quantum field theory isn't a subset of different "dynamical equations". Instead, it is another approach to define any quantum field theory via axioms etc. That's why it's a passion of mathematicians or extremely mathematically formally oriented physicists and one must add that according to almost all practicing particle physicists, they're obsolete and failed (related) approaches which really can't describe those quantum field theories that have become important. In particular, AQFTs of both types start with naive assumptions about the short-distance behavior of theories and aren't really compatible with renormalization and all the lessons physics has taught us about these things. Constructive QFTs are mainly tools to understand the relativistic invariance of a quantum field theory by a specific method.
Then there are many special quantum field theories, like the extremely important class of gauge theories etc. They have some dynamics including gauge fields: that's a classification according to the content. QFTs are often classified according to various symmetries (or their absence) which also constrain their dynamical laws: supersymmetric QFTs, gravitational QFTs based on general relativity, theories of supergravity which are QFTs that combine general relativity and supersymmetry, chiral QFTs which are left-right-asymmetric, relativistic QFTs (almost all QFTs that are being talked about in particle physics), lattice gauge theory (gauge theory where the spacetime is replaced by a discrete grid), and many others. Gauge theories may also be divided according to the fate of the gauge field to confining gauge theories, spontaneously broken QFTs, unbroken phases, and others. String field theory is a QFT with infinitely many fields which is designed to be physically equivalent to perturbative string theory in the same spacetime but it only works smoothly for open strings and only in the research of tachyon condensation, it has led to results that were not quite obtained by other general methods of string theory.
We also talk about effective quantum field theories which is an approach to interpret many (almost all) quantum field theories as an approximate theory to describe all phenomena at some distance scale (and all longer ones); one remains agnostic about the laws governing the short-distance physics. That's a different classification, one according to the interpretation. Effective field theories don't have to be predictive or consistent up to arbitrarily high energies; they may have a "cutoff energy" above which they break down.
It doesn't make much sense to spend too much time by learning dictionary definitions; one must actually learn some quantum field theory and then the relevance or irrelevance and meaning and mutual relationships between the "variants" become more clear. At any rate, it's not true that the classification into adjectives is as trivial as the list of colors, red, green, blue. The different adjectives look at the framework of quantum field theory from very different directions - symmetries that particular quantum field theories (defined with particular equations) respect; number of local excitations; ability to extend the theory to arbitrary length scales; ways to define (all of) them using a rigorous mathematical framework, and others.
Irving Segal, the obnoxious and disliked genius mathematician of MIT, once went around asking physicists « ¿what is a quantum field? » As he tells the story, only Enrico Fermi gave him an answer (after pausing for a little thought). « The occupation number formalism.»
What does this mean. What Fermi meant is that for a given particle, say an electron, there is a quantum field (in this case, the electron field. For a photon, it would be the electromagnetic field of Maxwell.) This particle could have many different states. Well, make a list of each possible state and put in the appropriate place in the list the number of particles in the Universe which are in that state, i.e., as we say, « occupy » that state. That is a quantum field. All the rest is sound and fury, ahem, all the rest is the attempt to study this concept mathematically, as prof. Motl explained very well. I wish this helped....
If I remember correctly, the beginning of Feynmans's notes on Quantum Electrodynamics, the first successful quantum field theory, give a short exposition of the occupation number formalism.
As Joseph says, Luboš touches everything in his usual way.
Less usefully than Luboš' Answer, but perhaps more specific for a mathematician, for something hypertopical you might try a Stonybrook workshop that ended yesterday on "Mathematical Foundations of Quantum Field Theory". http://scgp.stonybrook.edu/scientific/workshops/1498 has videos of talks by some of the players. The Buchholz PDF (no video in this case) looks a nice summary of AQFT at first glance, and Jaffe talks about a number of the variants you mention. You'll have to choose amongst the other talks according to your mathematical tastes, but the stated aim of the conference is
to review the current state of the field, agree on what has been accomplished and what could be accomplished by a systematic application of the known ideas and techniques, try to identify where new ideas and techniques could have the most impact, and agree on a list of important problems and questions whose resolution would at the least serve as benchmarks to measure our progress, and at best significantly advance the field.
Math papers usually leap into their own particular approach, citing a few papers to place themselves. I can't think of a paper that really taxonomizes and places all the approaches you mention relative to each other.
