I am currently unsure whether I want to do my PhD in psychology or neuroscience so I'm applying to both programs. The issue is that all of my previous research experience is exclusively in neuroscience. Within my psychology applications, do I need to explain why I only have neuroscience experience in my statement of purpose? If so, how should I go about doing this? I was a psychology major and I became interested in neuroscience during my second year, so that's why how I chose my first experience. I chose my second experience because they hired me due to my initial experience so I didn't have a lot of leeway with this one. I am still unsure which subject I prefer however.
You ask how you should go about applying. Just do it. The evaluation of whether your background is lacking or not will be made be any a committee at any institution you apply to. We can't judge that here. And, your record is what it is.
But the committee will e looking for indicators of success so make sure that your materials and statements indicate a high likelihood of success, not the possibility of failure. In particular use positive terms in describing your background. Don't say "Sorry, I only studied neuroscience...". Say, "I have a strong research background in neuroscience". That is to say, leave out words like "only".
I don't know the fields at all, but on the face of it I would think that (a) any research experience at all is a plus and (b) neuroscience gives you pretty good insight into some of the problems in psychology.
In some places there is an opportunity in doctoral studies to take additional courswork to fill in any gaps. This would be true in the US, for example.
Keep it positive. Knock 'em dead.
There are many fields in psychology that have a close relationship with neurosciences such as cognitive psychology, pharmacology, rehabilitative psychology etc. So it is worth exploring and explaining how your previous experience in neuroscience will enrich and support your future ambition in psychology.
You have done some psychology subjects by the sounds of it, so you do have some relevant experience. Just remember though, for most countries, there are strict regulation around clinical practice and becoming a registered psychologist. A research psychology PhD would not meet those requirements, so do your research.