I've just learnt about both types of NMR, and I haven't ever used NMR in real life, so anything I say may be wrong.
The crux of this question is that, in my own opinion, proton NMR gives you more information than carbon NMR, which I'd imagine to be much more helpful when analysing compounds. Because of this, can anyone tell me whether carbon NMR is ever used instead of proton NMR (which seems to be more 'powerful')? If it is, when?
TL;DR Yes, sometimes. However, in general 13C data is much less useful than 1H (and 2D) data and the truth is we can often do without it.
I will limit my answer to the context of organic chemistry. Firstly, I directly answer your question: is 13C NMR ever used instead of 1H NMR? The answer is, yes.
However, I can only think of one case where one would do so. This is when there is severe peak overlap in the 1H NMR, such that one cannot reliably use it to ascertain whether one has (for example) a mixture of compound, or just a single compound. In such a case the 13C NMR (which has much more dispersed peaks and hence nearly no overlap) is more informative than the 1H NMR. This is generally not very common.
I want to go beyond a direct answer, though, and talk about the usefulness of 13C data in general. You are correct about the 1H NMR being more helpful than the 13C. This is generally true, and the main reasons for this are twofold:
13C NMR simply does not give as much information as 1H NMR. The only useful thing that one can obtain from the 13C NMR is the list of chemical shifts – but in turn, these are virtually impossible to interpret for all but the simplest molecules. On the other hand, 1H NMR gives you information about proton environments (shifts), number of protons (integrals), and limited information about molecular connectivity (couplings).
As porphyrin wrote, 13C NMR takes significantly longer than 1H NMR, because (a) the natural abundance of 13C is much lower (b) the gyromagnetic ratio of 13C is smaller.
As such, out of all the NMR data that organic chemists collect, the 13C NMR is by far the least useful and most time-consuming spectrum. On its own, its only real use is to provide a "fingerprint" of the molecule, and perhaps some information about functional groups present. So, in most cases, the 13C NMR is indeed less helpful than the 1H NMR.
The question then becomes, "why do people still take 13C NMRs?" Before going on, though, I should make it clear that people do not take 13C NMRs preferentially over the 1H, but rather in conjunction with it.
It primarily comes back to the fingerprint idea – when everybody tabulates 13C shifts of molecules they make, this gives chemist A the information they need to figure out whether they were successful in making what chemist B previously made. You might argue that the 1H is good enough as a fingerprint, but this is not always true, and anyway the consensus is that two is better than one: all major journals require submission of 13C NMR data for novel synthesised compounds.
The only other real case where the absolute 13C shifts are useful are when one needs to resolve very close peaks in 2D spectra (e.g. HSQC, HMBC), or in compounds with very few protons (thus rendering H–H or H–C correlations in 2D spectra less useful). Even then, it is not true to say that the 13C data is preferred over the 1H data.
All in all, your question is perfectly justified. In most cases, the 13C NMR is less useful as the 1H NMR, and only in some limited cases is it as useful as the 1H NMR.
In fact, there is a pretty recent perspective article by Liermann and Schlörer[1] arguing that the 1D 13C NMR should not be recorded, except for in the special cases described in the previous paragraph. The authors also make the point that the 2D spectra serve as much more useful fingerprints and are also much better at validating a proposed structure for a molecule. Even though in crowded spectra the 13C fingerprint may be more useful than the 1H fingerprint, it's still less useful than the 2D data, where peaks are dispersed in two dimensions.
Reference
In principle the method is the same for both nuclei but there are practical differences. Proton NMR is approx 4 times more sensitive due to the difference in magnetogyric ratios but more importantly 99.98% of $^1$H are present in a molecule but only 1.1% of $^{13}$C. So unless you synthesise with $^{13}$C, (v expensive etc.) proton nmr is most commonly used. $^{13}$C can then be used as well if proton NMR is not definitive. Note also that NMR does not generally give you a structure (bond lengths, angles) just the topology, i.e. this atom next to that group etc.
Practical aspects are very important when deciding which experiments to run, and Proton 1Ds are by far the simplest and fastest experiment you can run for common organic molecules. 13C experiments take much more measurement time, that alone is a severe drawback. There might be exceptions, but in almost every case a Proton 1D is the first experiment you measure, and the only question is which experiment you run in addition to that one.
There are cases where the 1H experiments are not all that useful, for example if you molecule has very few Protons. You'd probably still run that experiment, just to make sure that there isn't other stuff in your sample that has more Protons.
But one important aspect is that those experiments give you different kinds of information. And then there are many more advanced NMR experiments that give you much, much more than those simple 1D spectra. What you measure depends on how difficult your problem is, and how sure you need to be that you're right. You can get a lot of information from a 1H 1D spectrum, but a lot of it is somewhat indirect. With 2D experiments like COSY, TOCSY, HSQC or HMBC you can directly measure the connectivity between nuclei in your molecule. Each experiment gives you different information, no single experiments gives you all the information about a molecule.
http://www.chem.ucla.edu/~harding/notes/notes_14C_cnmr.pdf
Take a look at this, for example.
When an organic chemist decide to make an NMR analysis, it is mostly to characterize a new synthesized molecule. And (s)/he will do both H1 NMR, C13 NMR, IR, MS and try to understand by looking all of them.
But if you still want to compare or only select one of them, H1 NMR will be much faster and in many cases gives many information, yet for instance when there is no hydrogen bonded to an atom, for especially bigger molecules, not just for the questions of organic chemistry or NMR courses, you will mostly not be able to see the entire molecule by merely H1 NMR. Well, again noone will guarantee that C13 NMR will make you completely solve the structure.
Funny it is, that even there are professors who say that looking solely on MS, if you know enough you will get the entire molecule in the end. In theory you can even say this to IR spectrum since neighboring atoms really have some effect on place and strength of absorption. However, these are some marginal thoughts, consumes very high amount of time if you try to apply and much prone to err compared to having as many analysis data as possible.
1H is probably the most common one, it is fast and it is quite intense for small quantities. You can use it as a fingerprint to compare results or you can determine at least some information from it. Hence it is preferred. But whenever we make a new compound or whenever we are unsure we also make a 13C. The problem is that you require more substance or a higher concentration or you won't be able to see much later. In 13C you cannot 'hide' your carbons as easily as in 1H. And it is sometimes useful because you can see if something is quarternary like in a substituted benzene ring. I think it is less used because it takes more time to measure and is often done during the night so the NMR will be available for others during the day. But that is just an operational factor so I wouldn't count that here. In general, although many groups do not do it when they publish their results, it is of interest to always mention the retention factor for the column chromatography or the boiling point under vacuum, the 1H NMR, the 13C NMR, the IR and the mass spectrum. 13C is also less used in practical courses at universities so students often believe it would be a problem. Same for 2D NMR. The option is always available and shouldn't be a problem if you ask the operator. But this common myth that it is something you wouldn't do because of costs or difficulty is just not true. If you cannot determine your compound you can always ask for the more advanced option. So in the end it depends, yeah 1H is standart because in most cases you know what you want to do but 13C is quite common as well.
