As mentioned above, there's absolutely no gain if you're not into the technical details.
In fact, there is a huge benefit (more below). But you need a tight workflow and especially a high-end sample-rate converter (which is not that easy to get). One simple mistake and you'll do more harm than good.
There are at least four good reasons for using higher sample-rates under certain circumstances:
1. The ADA conversion (chris mentioned it already). Especially in the case of 44.1kHz, the nyquist filter only has a quarter octave to filter 96dB (for 16bit quality) to 144dB (24bit) away! This is an extremely steep and most probably bad sounding processing. No matter if it's about recording or playback, higher resolutions tend to deliver better results because they substantially relax the nyquist filter.
I made a few graphs to visualize what happens in a AD/DA converter. Every frequency above Nyquist must be attenuated by at least ~96dB to deliver a dynamic range comparable to 16 bit. Here's how this looks like at different rates:
Pic 1: As you can see, a sample-rate of 44.1kHz asks for a crazy steep filter (a 351 taps FIR filter was required to create this response).
Pic 2: At 48kHz, the filter has more room to work, which is likely to introduce less negative side-effects (only 91 taps for this filter).
Pic 3: At 96kHz, the filter has an very easy job (only 19 taps for this filter).
2. Processing quality. Now this is where the "wow" is. I've already written countless posts about the topic, so a search might help. It's not that obvious, so I need a few lines...
The bandwidth of a digital system is clearly limited and defined by the sample-rate. The nyquist frequency is exactly about half the sample-rate (google for nyquist if this is a new concept for you, it's digital audio basics).
So, a samplerate of 40kHz results in an available bandwidth of 20kHz. But, there's an important detail here. While signals exceeding the bandwidth in an analogue system are simply attenuated, something radically different happens in its digital counter-part. The exceeding frequencies are instead mirrored back from the nyquist point into the audible spectrum (inverted of course).
Now, all non-linear processes are know to create new frequencies. Non-Linear processors are all kind of dynamics processing effects like compressors, saturators and clippers. As well as certain special effects like ring modulation, frequency modulation, gain modulation, exciters, transient designers, all kind of pitch/time stretcher and more. Most modern EQs are also non-linear because they all work with saturators to create pseudo analogue "color". Most of these new frequencies are new harmonics based on (and placed above) the signal (in fact, these are even and odd multiples of the signal's frequency).
This simply means that a say 8Khz tone being slightly saturated will create harmonics at 16kHz (the 2nd harmonic), 24kHz (the 3rd), 32kHz (the 4th) and so on, decaying to infinity.
But the bandwidth of the 40Khz samplerate system won't be able to carry the 3rd and 4th harmonic! These two harmonics mirror at 20Khz. The 3rd 4Khz above the limit, so its alias will land at 16kHz. The 4rth is 12Khz above the limit, so its alias will land at 8Khz. Of course, these aliases are now harmonically completely unrelated and make the saturation sound ugly. This is just an example, real life is much worse. BTW this is the main reason why high end analogue EQs and compressors are still unmatched. They don't have the unnatural bandwidth behaviour typical digital systems have.
A simple sine-sweep, a cheap saturator and a FFT visualizer will demonstrate it clearly (and also make the aliasing VERY audible).
Or, just watch a helicopter flight on TV, the rotation is faster than 24 fps (which is 12Hz), so it creates aliases that seem to move backward.
The benefits are huge in these cases. Not only do non-linear effects sound much much better at higher rates, even the cheapest synths deliver way better results at higher rates. But again, no proper sample rate convertion for the final jump to 44.1kHz and you fvck up everything.
3. Media requirements. Some DVDs and all Blue-Ray discs ask for a 24bit/96kHz resolution. You can use a lower sample-rate, but keep in mind how cheap DVD players are - their DA converters are of extremely low quality and built to work best with high resolution material (i.e. they use cost-efficient nyquist filters). See point 1.
4. Future developments. It may be useful to have a high resolution mix in your archive, even if the current technical consumer standard is lower. A future standard my ask for it.
Some AES papers mention a ~100kHz bandwidth requirement for impeccable high end digital audio processing(!). This is very interesting since the analogue world also has that rough "you need 100Khz of bandwidth for high end processing". This would be a sample-rate around 192kHz. Some radio processors like the Orban tools even oversample 16x for their clipping algorithm.
Ok, enough tech talk. Just work in 44.1kHz like you did before, but at least try it out and render your project at a higher rate. Reason for example allows to change the sample-rate on the fly, the before/after is shocking, really. Convert it down to 44.1kHz with a decent SRC later. You will hear a difference, it will sound much rounder, deeper, crispier and smoother at the same time.
Don't just think about your ears, also give the math some room to work. Even the cheapest EQs and synths start to excel when being oversampled (i.e. "using more samples the ear needs").
High end quality SRC recommendations:
Isotope 64bit SRC (delivered as bonus bundle with soundforge). Very nice, really. The Weiss Saracon is technically slightly better, but more expensive (Still very fair ~850$, I mean it's a Weiss product

).
If both above are too expressive or hard to find, check out Voxengo's R8Brain free, which is also of very high quality.
Finally, never ever trust the sample rate converters included in your DAW/Audio Editors. No matter if it's logic, PT, Cubase, Sonar, Wavelab, Soundforge, all have really bad SRCs built in.
Overview here:
http://src.infinitewave.ca/