Helium atoms are composed of a nucleus with 2 protons, 2 neutrons, as well as an electron shell with 2 electrons. The alpha particle consists of the nucleus alone. The fact they are written the same way is just an example of crappy notation in science.
The helium atoms are stable and inert. In other words, they don't go into chemical reactions with your body very easily and don't create any harmful byproducts.
Alpha particles on the other hand have a few different possibilities. Firstly they will look to pick up electrons to form a stable atom and hence they will tend to ionize at least a few atoms as they slow down to sufficiently low speeds. So if you ingest a horde of alpha particles then you start having a lot of electrons stripped from the different elements inside your esophagus and stomach and what not. You can imagine this is bad because then you're creating a lot of unstable configurations which could have more radioactive decay, for example, putting out beta particles.
If the alphas have high enough energy, they could actually knock the nucleus out of some other atoms, which then is again going to generate a lot of secondary radiation from those new unstable configurations. In this case, it would probably be a lot worse, but the same basic effect.