I would like to ask if you could kindly explain the difference between the big madd letters and the little madd letters.

Question
As-salaamu alaykum a rahmatullaahi a barakaatuhu

Jazaakumullaahu khayran for this excellent service- Maashaa'Allaah Tabarakalaah! I would like to ask if you could kindly explain the difference between the big madd letters and the little madd letters.  Why are some madd letters written as small letters and is the natural timing (madd Tab`ee) of the small letters different than that of the big letters in any of the modes of recitation?

Ahsaanallaah Ilaykum

Answer
Wa alaikum assalaam wa rahmatullahi wa barakaatuh,
Jazakum Allahu kahiran and baarak Allah feekum.
The medd letters written in a small form, such as the alif in are added by scholars and not part of the 'Uthmani writing.  Early Arabs knew that the alif was there, even though it wasn't written -for those who read this word with an alif-, but as time went on and more non-Arabs entered Islam the scholars of the mushaf added small letters to show that they are read, but at the same time, they did not disturb the original 'Uthmani writing of the mushaf. 
The same can be said of any small wow or ya' you may see, such as in or

There are times when a small medd letter, such as aayah 5 of al-Faatihah above is used for only some of the qira'aat and other modes of recitation do not read the word with the small alif.  Those who do not read the alif have no medd, just a fath-hah on the meem.  The natural medd, when read by any of the different qira'aat is always two vowel counts, but if they do not have a medd letter there, then they of course do not lengthen it.  The other mudood caused by a hamzah vary in length in the different ways of recitation. 

See also: http://www.abouttajweed.com/kb/entry/165/

Wa iyyaakum
Wa assalaam alaikum