The advantage is you don't need to know the channel characteristics ahead of time. In particular, if you used a Reed-Solomon code, you'd use one designed for one particular erasure rate. But that code would be too weak for a channel with a higher erasure rate (it would fail to decode) and it would be too strong for a channel with a lower erasure rate (it would be wasteful and use too many parity bits). The advantage of fountain codes is that they work efficiently whatever the channel's erasure rate.

By the way, I've been talking about channels with an erasure rate (the Binary Erasure Channel) because fountain codes were originally invented in that context. But you can find in the literature papers on "Rateless codes for noisy channels" that show that the fountain codes also work fine for bit-flipping channels like the binary symmetric channel and the additive white Gaussian noise channel.