What is the difference between concentration and mindfulness practice ? I am new to buddhism and meditation (I have been practising it for one month now), and despite finishing my fourth book on the matter, I am still very unclear about what a meditation session should look like when practicing either concentration or mindfulness. 

From what I understood, each of these practices are clearly differents and even have different pali names (Samatha and Vipassana if I am right ?).

I can already tell that I can't differenciate concentration and mindfulness.

When I focus on my breath, I am naturally distracted by all kind of things which I have to be mindfull of (the way my body feels, the sounds I ear, what I smell from the kitchen, my thoughts and emotions).

And when I try to practice mindfulness, I can't help but focusing on my breath, because it is often the first thing that I can notice in that state, and it is always present. 

When I start my meditation I focus on my breath, then if something happens I try to be mindfull of it, and then I focus on my breath again until something else happens, switching maybe every 2 to 10 seconds. 

I am wondering why concentration and mindfulness are referred to as disctinct practices in the books I read, as it doesn't make much sense to me. At this point I am afraid that I may be missing something important.
From what I understood, each of these practices are clearly different

Calm meditation deals with developing the Jhana factors which are:


Movement of the mind onto the object (vitakka)
Retention of the mind on the object (vicāra)
Joy (pīti)
Happiness (sukha)
One-pointedness (ekaggatā)


The object of calm can be conceptual object or one relating to ultimate realities (rupa, citta, cetasika and nirvana). If it is relating to ultimate realities then insight also develops.

Insight is based on meditation on the ultimate realities. The practice to understand the nature of the realities is through Satipatthana which are:


mindfulness of the body (kāya)
mindfulness of feelings or sensations (vedanā)
mindfulness of mind or consciousness (citta)
mindfulness of dhammās


Here you bring your mind in tune with the universal characteristics which results in wise attention. This helps you understand causality (Four Noble Truths, Dependent Arising, Conditional Relations):


  As such, bhikshus, the instructed noble disciple closely and wisely attends to dependent arising itself,
  thus:
  
  Imasmiṁ sati, idaṁ hoti; „When this is, that is;
  
  imass’uppādā, idam uppajjati. with the arising of this, that arises.
  
  Imasmiṁ asati idaṁ na hoti; When this is not, that is not;
  
  imassa nirodhā idaṁ nirujjhati. with the ending of this, that ends.‟
  
  Bhikshus, dependent on pleasant contact, a pleasant feeling arises.
  With the ending of the pleasant contact, the pleasant feeling that arose in dependent on that pleasant
  contact, ceases, is stilled.
  
  Bhikshus, dependent on painful contact, a painful feeling arises.
  With the ending of the painful contact, the painful feeling that arose in dependent on that painful
  contact, ceases, is stilled.
  
  Bhikshus, dependent on neutral contact, a neutral feeling arises.
  With the ending of the neutral contact, the neutral feeling that arose in dependent on that neutral
  contact, ceases, is stilled.


Assutava Sutta 2

Also wise attention is forerunner of cultivating the Noble Eightfold Path which leads to insight. Also it is one of the most essential ingredient in developing insight and insight meditation.


  Dawn, bhikshus, is the forerunner, the harbinger of sun-rise.
  Even so, bhikshus, for a monk this is the forerunner, the harbinger of the arising of the noble eightfold
  path, that is, accomplishment in wise attention.
  
  Bhikshus, when a monk is accomplished in this wise attention, it is to be expected that he will
  cultivate the noble eightfold path, develop the noble eightfold path.


Yoniso Manasikāra Sampadā Sutta

Insight  practice can be combined with calmness practice as in Saṅkhitta Dhamma Sutta:


  When, bhikshu, this samadhi has been cultivated, well cultivated by you, then you should
  train yourself thus:
  
  "I will dwell exertive, clearly aware, mindful,
  observing [contemplating] ________ the in the _________ [body | feeling | mind | dhamma],
  removing covetousness and displeasure [discontent] in regard to the world."
  
  Thus, bhikshu, you should train yourself.
  
  When, bhikshu, this samadhi has been cultivated, well cultivated by you, then, you, bhikshu,
  
  THE 1ST DHYANA:
  
  should cultivate this samadhi with initial application, with sustained application;
  
  should cultivate this samadhi without initial application, with only sustained application;
  
  THE 2ND DHYANA:
  
  should cultivate this samadhi without initial application, without sustained application;
  
  should cultivate this samadhi with zest;
  
  THE 3RD DHYANA:
  
  should cultivate this samadhi zest-free;
  
  should cultivate this samadhi attended by comfort;
  
  THE 4TH DHYANA:
  
  should cultivate this samadhi attended by equanimity


Both are needed for liberation.


  And what, bhikshus, is the path leading to the unconditioned?
  
  Calm and insight [Samatha and vipassana]—this is called the path leading to the unconditioned.


(Mūla) Samatha Vipassanā Sutta

Both calm and insight is needed to get to the final goal. They may not develop at the same space or level at the same time.


  (1) “insight preceded by calm” samatha,pubb’angama vipassanā
  
  (2) “calm preceded by insight” vipassana,pubb’angama samatha
  
  (3) “calm coupled with insight” samatha,vipassana,yuga.naddha
  
  (4) “a mind seized by dharma-restlessness” dhamm’uddhacca,viggahita manasa 


(Yuga,naddha) Paṭipadā Sutta

Also the development of both gives arise to psychic powers as in 


  In that case, Vaccha, cultivate two higher teachings: calm and insight. Vaccha, when these two
  teachings—calm and insight—are cultivated further, they bring about a penetration of the many elements
  (dhātu).
  
  ...


Mahā Vaccha,gotta Sutta

When one develops without the other then the following advice is given in finding a suitable teacher / technique:


  (1) A meditator who is able to attain only calm should consult an insight-attainer.
  
  (2) A meditator who is able to attain only insight should consult a calm-attainer.
  
  (3) A meditator who is unable to attain both should consult one skilled in both.
  
  (4) A meditator who is able to attain both should work for awakening.


(Samatha Vipassanā) Samādhi Sutta 3

When I focus on my breath, I am naturally distracted by all kind of things

To get over distraction you have to practice


Movement of the mind onto the object (vitakka)
Retention of the mind on the object (vicāra)


More effort and energy you put into it lesser distraction you will get from external distraction going forward.

Being mightful about the breath should be inline with the techniques in Anapanasati Sutta to get the benefits.

Also if your mind gets distracted then you have to:


know where you mind wanders away
each new object or through has a sensation of pleasantness, unpleasantness or neutral sensation associated with it. Get to know this sensation and wait for it to subside otherwise this itself and memories it trigger means the distractions will continue.
Bring back to the object of meditation 


And when I try to practice mindfulness, I can't help but focusing on my breath

Mindfulness of the breath is also insight and Satipatthana meditation.


First Tetrad: Contemplation of the Body (kāya)


Discerning the in and out breathing
Discerning long or short breaths
Experiencing the whole body (sabbakāaya)
Calming bodily formations

Second Tetrad: Contemplation of the Feeling (vedanā)


Being sensitive to rapture (pīti)
Being sensitive to pleasure (sukha)
Being sensitive to mental fabrication (citta-saṃskāra)
Calming mental fabrication

Third Tetrad: Contemplation of the Mind (citta)


Being sensitive to the mind
Satisfying the mind
Steadying the mind
Releasing the mind

Fourth Tetrad: Contemplation of the Mental Objects (dhammā)


Dwelling on impermanence
Dwelling on dispassion
Dwelling on cessation
Dwelling on relinquishment



Source: Anapanasati Sutta

But having said this ...


Movement of the mind onto the object (vitakka)
Retention of the mind on the object (vicāra)


... will help you keep your focus on a chosen object, may it be the breath or one of the Satipatthana.

I am wondering why concentration and mindfulness are referred to as distinct practices

The techniques are different but should be practiced together as highlighted above.

Also see: Samatha and Vipassanā by Piya Tan.