Why doesn't the human skin grow back exactly how it was before being damaged?
Human skin is made up of three layers [1]:


Epidermis; it contains no blood vessels; it has 5 cellular layers (starting from surface):


stratum corneum
stratum lucidum
stratum granulosum
stratum spinosum
stratum germinativum

Dermis; it is separated from the above by the basal membrane; made up of connective tissue (with blood vessels and nerve endings)
Hypodermis - it is not even considered part of the skin


"Gray940". Licensed under Public domain via Wikimedia Commons.


  Why (and how) is regenerated skin different from original ?


Why? Skin is a very important organ. It protects from pathogens, it maintains body temperature, it keeps body water [1], etc. How? That depends on what causes the regeneration. I've emphasized two terms in the above list.


Stratum corneum represents the "skin cells that becomes the dust in our homes and it always grows back to full thickness". It is lost during a physiological process called desquamation [2]. The cells that are lost are non living corneocytes. Their only purpose is to from a protection barrier [3].
Basement membrane holds the germinativum layer of the epidermis. Injuries that do not affect it are repaired by complete regeneration of all destroyed layers. However, if the injury destroys membrane continuity, the germinative cells on the edges will fail to "reconnect" with each other, thus the layers derived from them will follow the same rule. A process of fibrosis is triggered. The result of  extensive fibrosis is a scar, containing collagen secreted by the granulation tissue with dermal origins. Yet, the germinative layer tends to join back together [6], but probably the excess collagen production, without the limiting basement membrane, bulges into the epidermis and limits germinative layer extension. That's why keeping the edges of the tissue close to each other reduces excessive fibrosis and allows germinative layer regeneration, thus resulting in none to small, barely visible scars.


How deep should an injury be in order not to affect basement membrane? No more than 0.06 to 0.08 mm [5].


  What about skin transplants ?


Skin grafts contain at least a part of dermis [4], which will allow granulation tissue formation thus permitting revascularisation and making the graft stick. Otherwise it would fall off and/or be lost by necrosis.



References:


Wikipedia contributors, "Skin," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Skin&amp;oldid=619454074 (accessed August 4, 2014).
Wikipedia contributors, "Desquamation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Desquamation&amp;oldid=614028888 (accessed August 4, 2014).
Wikipedia contributors, "Stratum corneum," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Stratum_corneum&amp;oldid=615247326 (accessed August 4, 2014).
Wikipedia contributors, "Skin grafting," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Skin_grafting&amp;oldid=603324006 (accessed August 4, 2014).
Sandby-Møller J, Poulsen T, Wulf HC. Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits. Acta Derm. Venereol. 2003;83(6):410-3. doi: 10.1080/00015550310015419. PubMed PMID: 14690333. 
Dawn A. Tamarkin, Ph.D. Repair of skin. 2011 STCC Foundation Press. Available from http://faculty.stcc.edu/AandP/AP/AP1pages/Units1to4/skin/repairof.htm (accessed 04.08.2014)