How should I interpret this result in deciding what model to use for subsequent analyses.
Essentially the AIC offers a way to compare how much an increase in a model's likelihood is "worth" an increase in the model's complexity when determining model quality. Since you propose two models with similar AIC's what you should interpret is that according to this measure of likelihood vs. complexity trade-off they are essentially equal in quality.
Does it make more sense to use the simpler model because it is "just as good" as the full model and simpler
Not really as it is only "just as good" when simplicity has already been taken into account.
or to use the full model because it's not worse than the simpler model after model complexity is taken into account?
Again same answer but in reverse. The gain in simplicity is approximately balanced out by the difference in likelihood.
Does it not matter at all?
Yes and no. They are two different models and from a frequentist point of view there is only one "correct" model which describes the data, which could even in fact (and probably more likely) be neither of these. So in this sense it does matter which model you choose. But determining this through AIC shows choosing one or the other does not change the quality by any meaningful amount. Essentially do you prefer a model with greater likelihood to fit the data but with more complexity or vice versa, where the trade-off according to Akaike is essentially equal.
Or should I continue to investigate both models?
Do you expect to gain different but still meaningful insights through both models? Refer to @Bjorn