What are the differences in using an eyepiece with a barlow instead of a shorter eyepiece? I intend to start making visual observations, and I want to know more about the difference between using barlow lenses and one or two eyepieces and using a complete kit of eyepieces, both for planetary and deep space objects.
 A: In general, more glass means the image gets worse. However, if the barlow is good, the losses and aberrations are below the annoyance threshold or, rarely, below even the perceptual threshold. It's all in the eye of the beholder anyway.
Also, especially with cheaper eyepieces, it is sometimes the case that the more powerful eyepieces have less eye relief, while less powerful ones have more eye relief. But if you combine a less powerful eyepiece with a barlow, you preserve the long eye relief and you get more power. That's not the case if you have a set of high-end glass (which may provide plenty of eye relief anyway), but it's a useful trick otherwise.
Finally, in case you don't know this already, you can't just increase the magnification ad infinitum. Beyond a certain threshold, you just get large fuzzy blobs. Measure the aperture in mm, double it, and that's pretty much the threshold (it's not a hard limit, there is some variation both down and up). E.g., a 4" scope, that's a 100 mm aperture, which means don't expect much beyond 200x - and that's provided that all optics involved are perfect and the scope is very well collimated, which is rarely the case; otherwise, shave off some percentage points from it.
Other than that, no, there aren't many differences.
EDIT: Oh yeah, if your scope is short, let's say f/5 or less, cheap eyepieces will have trouble with the steep light cone and will distort the image at the edge. A 2x barlow makes an f/5 into an equivalent f/10 which is much easier on the eyepiece. But, OTOH, if your eyepiece is cheap enough that it can't handle an f/5, then your barlow is not high-performance either, so it may introduce some issues of its own. But usually there might be a gain if your scope is really short.
A: Modern Barlow lenses with modern glass and coatings have minimal effect on the quality of the image.  A good quality Barlow (don't bother with a cheap one!) will be an excellent addition to a beginner's hardware collection because it immediately doubles the range of magnifications available.
That said, you will find that most experienced observers very rarely if ever use Barlow lenses.  The main reason is simplicity.  Single eyepieces, even modern complex designs, are just a lot easier and safer to handle in the dark. Aside from giving superior optical results, they are far less likely to fall apart and come crashing to the ground.  Whenever I use a Barlow, my tension level increases noticeably.
Most experienced amateur astronomers have a Barlow lens or two kicking around, but they spend most of their lives in the eyepiece case.  The only times they get put into use are on that rare, once a year, night of perfect seeing, or if extra focal length is needed for planetary imaging. And even then, it will be a top end lens like a Tele Vue Powermate, which has minimal effect on the image.
