Some Arduino starter-kit projects show the resistor placed before the anode of a LED, coming from output, and others show it after the cathode, going to the ground. Is there any difference? Why?
There's no difference.
One of the basic principles of electric current is that the same current flows through all elements that are arranged in series.
The purpose of the resistor is to limit the current, it doesn't matter in which part of the circuit it is located.
Usually as RedGrittyBrick said it doesn't matter. You have non-reactive components (ideally, which is pretty close to reality) thus any current flowing into the diode/resistor must flow out, thus current would be limited equally well by having the resistor on either the anode or cathode side.
However, there are designs for driving LED's using transistors where choosing where you place your resistor does matter. For example, sometimes people will use a BJT transistor to allow for higher current drives through the LED because micro controller pins are limited to low current outputs. By cleverly placing the resistor it can serve multiple functions at the same time, thus allowing you to reduce the part count and save some money/space.
In the above schematic we are using the resistor R1 to both limit current flowing through the LED as well as limiting the base current (current from MCU output to R1). If we instead moved R1 above Q1 or even above D1, we would have to add an additional resistor between MCU output and Q1 to limit the base current. This is an extra part we would have to have in our design meaning extra cost and space.
It's unclear from the original question if the kit/tutorials you're using have this type of LED driving circuit. It's not unreasonable because a typical small LED's current is below the limit of being safe to drive directly from the MCU, but is slightly worry-some if you have a lot of other stuff going on. Just be aware that sometimes the arrangement of parts does matter, and other times it doesn't.
Is the cost significant? It really depends. Resistors are really cheap (unless you have super high requirements), and a resistor generally doesn't take up a lot of space. It also gives you better control over the two different currents, but it isn't free.