They generally both have a belief in multiple gods (polytheism), and are mostly people who do not acknowledge the Bible. This would apply to modern-day Heathens, but not necessarily apply to modern-day Pagans. Paganism can be seen as an umbrella term, with a wide number of religious beliefs and practices from around the world under it. From the Christo-Wiccan, to the Sikh, there are a multitude of practices and traditions that can be called Pagan.
In its simplest form, historically Paganism tends to refer to the classical religions (Roman, Greek etc) where Heathenry tends to refer to the old Germanic/Northern Europeans.
As to similarities it depends who you ask. If you ask a non-heathen, then a pagan is a heathen and heathen is a pagan. But if you ask someone say, a follower of Asatru or Druidism then they will probably say they are not of the Pagan tree.
So, while Heathenry is, technically speaking, a branch of the Pagan tree, many modern-day Heathens prefer to look at it as a separate, though a still-related tree within itself. A bit like the many factions of most world religions.