Almond paste and marzipan usually differ with the amount of sugar in the mixture. Using almond paste, you may end up with a less sweet cookie. Alternatively, you could experiment with adding extra sugar if you want to maintain the sweetness.
Almond paste will contain more almonds and less sugar, and the texture will be different than marzipan. Marzipan is made with more finely ground almonds and more sugar, which is why it molds so well. That difference in texture means they won't behave the same way in the recipe. That said, I've been stuck and done it myself. You will want to add in a bit more sugar, and I would probably give the butter, sugar and almond paste a good whirl in the food processor if at all possible, to help smooth things out. That only applies to almond paste and marzipan produced here in the US. If you are in Europe, or have a European product, then you can likely use it without concern. In Europe, the two items are usually not any different to one another. The major challenge is to get the texture smooth, and well mixed, so you don't end up with lumps or a gritty dough.