Generally speaking, you can use the adverb "recently" in both the past simple and the present perfect, without any difference in meaning. The use of this adverb in the past is more common in AE while in the present perfect is more common in BE. However, if you look at these minutely, there is sure some difference.
When you use the present perfect, you look at something as happening in the past but having a result in the present. In other words, we talk about the past and the present when we use the present perfect while, in the past simple, we look at an event as happening in the past only. For example:
I have purchased that CD recently indicates that I own or have that CD now.
On the other hand, "I purchased that CD recently" doesn't necessarily mean that I own or have it now. Maybe I sold it or it got stolen.