Differences between neurotransmitters and neuromodulators
Key points:


Neurotransmitters directly stimulate or inhibit ionotropic receptors (the receptors of ion channels) in the postsynaptic membrane within a single synapse. 
Neuromodulators act on metabotropic receptors within or outside the synapses of many neurons at the same time; they modulate the release of neurotransmitters and excitability of ionotropic receptors; the signaling between metaboreceptors and ionoreceptors usually involves a cascade of secondary messengers. 


Neurotransmitters

"Classical neurotransmitters" stimulate or inhibit the fast-acting ionotropic receptors of ion channels. Their actions last few milliseconds and are limited to a single synapse, which means they are released from a presynaptic membrane of one neuron and they target the postsynaptic membrane of one (or, sometimes two or maybe more) postsynaptic membranes of the neurons involved in the same synapse, or a single muscle cell or other target cell (Oxford Scholarship, The Revisionist).

Examples of neurotransmitters: noradrenaline, acetylcholine, dopamine.

Neuromodulators

Neuromodulators act on the slow-acting metabotropic receptors inside or outside the synapse of "thousands" of neurons at the same time. In long-term (minutes), they modulate the release of neurotransmitters and excitability of ionotropic receptors. They are usually (but not necessary) released from presynaptic membranes into the synapse by the same mechanisms as classical neurotransmitters but can then leave the synapse through "spillover" and act on the neurons other than those involved in the synapse. So, their action is diffuse, but usually still limited to to certain neural pathways. (International Encyclopedia of the Social &amp; Behavioral Sciences, 2001, Encyclopaedia Britannica, Wikipedia, Current Opinion in Neurobiology)

Examples of neuromodulators: dopamine (D receptors), serotonin (5HT receptors), acetylcholine (M and nicotinic receptors), noradrenaline (alpha and beta receptors), histamine (H receptors), endorphins and neuropeptides (Wikipedia, Journal of Chemical Neuroanatomy, 2011.) 

Neurohormones are released from neurons into the bloodstream and have a systemic effect, for example:


Oxytocin and vasopressin, which are produced in the hypothalamus travel to neurohypophysis from where they are secreted into the blood and act on the mammary glands and uterus. 
Adrenaline, which is secreted from adrenal medulla acts, via sympathetic nervous system on various tissues, usually on the smooth muscles of the blood vessels, heart, gut, etc. 


Can a certain molecule can be a neurotransmitter, neuoromodulator and neurohormone at the same time?

Yes, for example:

Noradrenaline acts as a classical neurotransmitter and neuromodulator in the postganglionic neurons of the sympathetic nervous system and as a neurohormone secreted from the adrenal medulla (International Encyclopedia of the Social &amp; Behavioral Sciences, 2015)

Dopamine acts as a classical neurotransmitter and neuromodulator in the dopaminergic pathways in the deep areas of the brain, such as thalamus and pituitary gland (picture); as a hormone, it inhibits the secretion of prolactin from the pituitary gland.

Are both neurotransmitters and neuromodulators stored in and released from vesicles by similar mechanisms?

Yes or no. Neurotransmitters are always released into the synapse, while neuromodulators may or may not be released into the synapse (Oxford Scholarship).

Do both neurotransmitters and neuromodulators get effective by interacting with receptors and affect if a ligand-gated ion channel opens or not?

Neurotransmitters can be effective if the ligand-gated ion channel are open and neuromodulators can open or close the ion channels.



Since the same molecule can be both a neurotransmitter and neuromodulator, it may not be that important to define the differences between the both in detail:


  It could, therefore, be concluded that the aspect of defining
  neurotransmitters has taken a back seat to the investigation of the
  actions of individual signaling molecules. (International Encyclopedia of the Social &amp; Behavioral Sciences, 2015)