What's the difference between the neuroendocrine system vs endocrine system?

Neuroendocrine systems can be defined as the sets of neurons, glands and non-endocrine tissues, and the neurochemicals, hormones, and humoral signals they produce and receive, that function in an integrated manner to collectively regulate a physiological or behavioral state.

The neuroendocrine system is the mechanism by which the hypothalamus maintains homeostasis, regulating reproduction, metabolism, eating and drinking behaviour, energy utilization, osmolarity and blood pressure. (Source)

The central neuroendocrine systems serve as an interface between the brain and many of the peripheral endocrine systems. This chapter discusses the hypothalamic control of anterior pituitary systems regulating stress, basal metabolism, growth, reproduction, and lactation. Each of these systems involves one or more hypothalamic releasing or inhibiting hormones, released from hypothalamic neurons that terminate in the portal capillary vasculature that projects from the median eminence at the base of the hypothalamus to the anterior pituitary gland. There, the hypothalamic hormones act upon subsets of anterior pituitary cells to regulate pituitary hormone release and downstream physiological functions. Other hypothalamic neuroendocrine cells control water/salt balance, and lactation/parturition, through the release of vasopressin and oxytocin from nerve terminals that arise in hypothalamus and project to the posterior pituitary gland. Together, these hypothalamic neuroendocrine functions enable the central nervous system to respond rapidly to internal or external environmental change, and to maintain a response through endocrine hormonal transducers. (Gore, 2013).

The endocrine system is a group of glands and other structures that release internal secretions called hormones into the circulatory system.

Endocrine organs are richly vascularized ductless organs that produce hormones. The epithelial cells of the organ secrete their hormone product directly into the bloodstream where, upon binding with specific receptors in target organs, cellular functions are affected. The endocrine systemincludes not only the endocrine glands (pineal, pituitary, thyroid, parathyroid, and adrenal) but also single cells and small clusters of cells in the thorax and abdomen known as paraganglia. This chapter addresses the thyroid, parathyroid, adrenal, pituitary and pineal glands, and paraganglia. In addition to size differences, several features of endocrine tissues differ between rodents and humans, and often between rodents. (La Perle & Dintzis, 2018).

Question 1

When it comes to the brain, are the neurosecretory cells located in the hypothalamus or the posterior pituitary gland or both?

Neurosecretory cells are found in the hypothalamus. (Source)

There are two types of neurosecretory cells: magnocellular cells and parvocellular cells. Magnocellular cellsare large cells, having long axons that terminate in the neurohypophysis (also known as the posterior pituitary). Magnocellular cells release the hormones vasopressin and oxytocin to the general circulation; we will talk about the physiology of these hormones in Conjoint 403.  The parvocellular cells are small cells, with shorter axons that terminate at a capillary-rich bulge at the base of the hypothalamus known as the median eminence. 

Question 2

Is the hypothalamus and the posterior pituitary gland classified as neuroendocrine glands...

No

...and is the anterior pituitary classified as an endocrine gland?

Yes.

The following is from: https://courses.lumenlearning.com/boundless-ap/chapter/the-pituitary-gland

(emphasis mine)

The posterior pituitary (or neurohypophysis) comprises the posterior lobe of the pituitary gland and is part of the endocrine system. Despite its name, the posterior pituitary gland is not a true gland; rather, it is largely a collection of axonal projections from the hypothalamus that terminate behind the anterior pituitary gland.

The pituitary gland is an endocrine gland, about the size of a pea, that sits in a small, bony cavity at the base of the brain whose secretions control the other endocrine glands and influence growth, metabolism, and maturation.

While the pituitary gland is known as the master endocrine gland, both of its lobes are under the control of the hypothalamus: the anterior pituitary receives its signals from the parvocellular neurons, and the posterior pituitary receives its signals from the magnocellular neurons.

The anterior pituitary, also called the adenohypophysis, is a major organ of the endocrine system, and is the glandular, anterior lobe of the pituitary gland.

The anterior pituitary gland secretes 7 hormones: follicle stimulating hormone, luteinizing horomone, adrenocorticotropic horomone, thyroid stimulating horomone, prolactin, endorphins, and growth hormone.

The posterior pituitary gland does not synthesize any hormones. Hormones known as posterior pituitary hormones are synthesized by the hypothalamus, and include oxytocin and antidiuretic hormone (ADH), also called arginine vasopressin or vasopressin (Liu, 2007). The hormones are then stored in neurosecretory vesicles (Herring bodies) before being secreted by the posterior pituitary into the bloodstream.

With regard to neurohormones mentioned in question 3, neurohormones are hormones that are produced by neurosecretory cells and released by nerve impulses (e.g., norepinephrine, oxytocin, vasopressin).

This is the part which can be confusing. Whilst ADH and oxytocin are released from the posterior pituitary gland, which is an endocrine gland, the hormones are produced by neurosecretory cells within the hypothalamus and they are stored in and released from neurosecretory vesicles within the posterior pituitary gland. Therefore they are classed as neurohormones.

References

Gore, A. C. (2013). Neuroendocrine systems. In Fundamental Neuroscience (Fourth Edition) (pp. 799-817). doi: 10.1016/B978-0-12-385870-2.00038-X

La Perle, K. M. D., & Dintzis, S. M. (2018). Endocrine system. In Comparative Anatomy and Histology (pp. 251-273). Academic Press. doi: 10.1016/B978-0-12-415759-0.00058-3

Liu, K. D. (2007). Hyponatremia and Hypernatremia. In Critical Care Secrets (Fourth Edition) (pp. 285-290). doi: 10.1016/B978-1-4160-3206-9.10044-8

Not all hormones enter the blood - just as a quick addendum to the thorough reply above, and as per a comment above stating not all hormones from neuroendocrine cells go into circulation - seems to be true: many of the neuroendocrine cells release hormones directly into the organ where those cells appear - as per the gut, heart, lungs - here's a nice visual overview

https://byjus.com/biology/hormones-heart-kidney-git/