To understand the difference between a confidence interval and a prediction interval you need to understand what each is trying to do.
Imagine you have a target population you would like to learn something about. This population could be patients at a large hospital.
You conduct a study in which you select 100 patients at random from this target population. For each patient in your random sample, you measure their whole blood hemoglobin concentration (to stay with the example you mentioned). These measurements will become your data
Your first study goal will be to estimate the average value of the whole blood hemoglobin concentrations for ALL patients in the target population. To this end, you will compute a 95% confidence interval for that average. This interval will give you the likely range of values of this average. (Another word for average is mean.)
Your second study goal will be to predict the whole blood hemoglobin concentration for A NEW randomly selected patient from the target population. To this end, you will compute a 95% prediction interval. (This would be patient # 101, which was not part of your original sample of 100 randomly selected patients from the target population.)
Both intervals are computed from the same data, but are used to guess different unknown values.
The 95% confidence interval will help you guess the average value of what you are interested in - whole blood hemoglobin concentration - for ALL the patients in your target population of patients.
The 95% prediction interval will help you guess the value of what you are interested in - whole blood hemoglobin concentration - for a single patient in your target population of patients, randomly selected from the target population but originally not included in your study sample of 100 patients.
Both of these intervals have long-run properties. If you repeated your study a large number of time, each time selecting a different sample of 100 randomly selected patients from the target population and computing your 95% confidence interval and 95% prediction interval following the same statistical methodology, you would expect that:
95% of the confidence intervals will include the average value of whole blood hemoglobin concentration for ALL the patients in your target population of patients;
95% of the prediction intervals will include the value of whole blood hemoglobin concentration for a new, randomly selected patient in your target population of patients.