When we exercise and we need more air, we open our mouths and indeed it is easier to breathe. When we sleep, however, the situation is totally different; breathing through the nose is a necessity, not an option.
In research on normal patients, the upper airway resistance (and therefore the effort required to breathe) is much higher with mouth breathing compared to nose breathing.
Why is this the case? The main reason is that when we sleep, the muscles of the throat and tongue are relaxed (especially during rapid eye movement -REM- sleep). In patients who sleep with the mouth open because of blocked noses, the walls of the throat tend to collapse backwards even more, causing further narrowing of the upper airway.
Thus nasal obstruction can be a major reason for snoring and/or obstructive sleep apnoea (OSA) 
. Since OSA increases the risk of high blood pressure, heart disease and cerebrovascular disease
(stroke), mouth breathing can indirectly be harmful to a person's health.
For children, nasal obstruction is usually caused by enlarged adenoids and /or inferior turbinates (usually due to allergic rhinitis). It is well known that such children tend to develop the so-called "adenoid facies
- open mouth, narrow midface and long lower jaw.
One of the main features of this is maxillary hypoplasia (decreased growth of the jawbones), which is usually associated with crowding of the teeth and dental malocclusion (crooked teeth that do not fit).
These changes occur during the period of growth of the face which roughly correlates to when one's adult teeth are developing.
Because nasal obstruction 
indirectly leads to developmental abnormalities of the jaw and teeth, treating nasal obstruction and prevention of mouth breathing at this crucial age can help reduce the development of these conditions, which would otherwise need orthodontic and orthognathic treatment (e.g. jaw surgery, braces).
Furthermore, while many factors may contribute to OSA in adults, leaving nasal obstruction untreated in a child may set him/her up to have narrow upper airways with less reserve to accommodate the narrowing of the upper airway from other factors (eg weight gain, decreased muscle tone with age).
If your nose feels blocked while you are awake, it is more than likely that it is blocked when you are asleep. Therefore, it is important to have your nose evaluated and treated so as to minimise the downstream health risks.