What exactly is an FPGA?
An FPGA is not like a GPU. A GPU is a pre-designed processor with all the circuitry already there. All you have to do is hook it up and let software interface with it. FPGAs, on the other hand, are completely different beasts. They are a "blank slate" processor. You have to design an FPGA, not just buy one. You determine what transistors do what, how they hook up, etc. They are efficient because every part has a job, but they require you have a deep understanding of electronics and integrated circuit design.
In the question you linked explaining what makes a GPU so good at password cracking, it says that a GPU has a large number of identical cores that can run the same algorithm in parallel. How the core works internally is something that you don't need to know in order to make use of it (and likely can't legally know). For an FPGA, however, there is no such unit. You create each core and design it. You set up the individual logic gates. Basically, you design the chip, using the FPGA as nothing more than the actual, physical hardware that contains the transistors. Developing on them can be difficult.
Which is faster for hash cracking?
For the same price, a high-end GPU is likely to be faster than an FPGA at processing hashes, but an FPGA is likely to use less power for processing the same number of hashes. It's important to note that there are several different kinds of FPGA. An FPGA "conversion" involves creating an ASIC from FPGA HDL. This is often quite expensive and is only done when purchasing in quantity, but tends to be a bit faster than a regular FPGA (apparently around 30% faster), and much easier, and cheaper, than designing an ASIC. A true ASIC is, of course, much more efficient than either. Designing an ASIC is even more difficult than programming an FPGA, and requires far more knowledge of low-level integrated circuit design. Overall, ASICs are efficient and fast, but costly and hard to design.
Which should you get?
If you want the highest possible MHash/s for a given price, or if you need the flexibility of supporting multiple algorithms, get a GPU. If you want the highest possible MHash/J (joule, an energy unit), get an FPGA. If you want both speed and efficiency and do not care about price, design an ASIC.
If you are new to password cracking and just want to build an efficient cracking rig, get a few fast GPUs. FPGAs and ASICs are typically used if you are a company with a team of developers with a large (but strict) budget to work with, when custom solutions are needed. For most password cracking needs, a GPU would do just fine, not only because of their competitive speed, but because popular hash-cracking software like Hashcat is designed to make use of GPUs for acceleration. Nothing of that sort exists for FPGAs or ASICs, meaning you'd have to design it all yourself.
An excellent post on the Electronics Stack Exchange explains the difference between FPGAs, ASICs, and general-purpose microcontrollers (which a GPU is most similar to) in more detail, from an electronics design perspective, rather than a hash-cracking perspective.