Assalaamu `alaykum
With regards to the evidence for reciting Salawaat after the final tashahhud, then this is the generally what is quoted as evidence:
عَنْ أَبِي مَسْعُودٍ الأنصاري رضي الله عنه قَالَ: قَالَ بَشِيرُ بْنُ سَعْدٍ: يَا رَسُولَ اللهِ! أَمَرَنَا اللهُ أَنْ نُصَلِّيَ عَلَيْكَ، فَكَيْفَ نُصَلِّي عَلَيْكَ؟ فَسَكَتَ، ثمَّ قَالَ: «قُولُوا: اللّهُمَّ صَلِّ عَلَى مُحَمَّدٍ، وَعَلَى آلِ مُحَمَّدٍ، كَمَا صَلّيْتَ عَلَى إبْرَاهِيمَ، وَبَارِكْ عَلَى مُحَمَّدٍ، وَعَلَى آلِ مُحَمَّدٍ، كَمَا بَارَكْتَ عَلَى إبْرَاهِيمَ فِي الْعَالَمِينَ إنَّكَ حَمِيدٌ مَجِيدٌ. وَالسَّلاَمُ كَمَا عَلِمْتُمْ». رَوَاهُ مُسْلِمٌ.
Ibn Mas`ood al-Ansaaree (May Allaah be pleased with him) said: Basheer ibn Sa`eed said: “O Messenger of Allaah, Allaah ordered us that we send prayers upon you. So how do we send prayers upon you. He (the Messenger of Allaah :saw2: ) remained silent. After a short while, he :saw2: said: “Say:
اللّهُمَّ صَلِّ عَلَى مُحَمَّدٍ، وَعَلَى آلِ مُحَمَّدٍ، كَمَا صَلّيْتَ عَلَى إبْرَاهِيمَ، وَبَارِكْ عَلَى مُحَمَّدٍ، وَعَلَى آلِ مُحَمَّدٍ، كَمَا بَارَكْتَ عَلَى إبْرَاهِيمَ فِي الْعَالَمِينَ إنَّكَ حَمِيدٌ مَجِيدٌ
And then make Salaam (tasleem) as I have taught you” [Narrated by Muslim]
With regards to the opinions on this...
Then first is that reciting the Salawaat after the final Tashahhud is waajib. This is the opinion of the Shaafi`iyyah, the known opinion of the Hanaabilah (based on a narration from Imaam Ahmad) and from the contemporary scholars, Shaykh Ibn Baaz held this view.
The reason for them holding this view is because it mentions in the hadeeth that the Messenger of Allaah :saw2: said: قُولُوا which means "Say". The scholars mentioned that this indicates that it is an 'Amr (a command) and from Usool al-Fiqh, we know that a command conveys obligation unless there is another evidence to indicate that it is not obligatory. Based on this, they mentioned it is Waajib to recite the salawaat.
The other opinion is that it is Mustahabb to recite the Salawaat after the final Tashahhud. This is the opinion of Imaam Maalik, Abu Haneefah and one report from Imaam Ahmad. This is also the view of the Jumhoor (majority of the scholars) as mentioned by Imaam al-Nawawi. One of their reasons was because, they mentioned that the Prophet :saw2: did not, himself, go and tell the Sahaabah to recite the Salawaat after Tashahhud, but rather he :saw2: was asked a question and he replied to the question. Because, if it was an obligation, he :saw2: would have mentioned it without even the Sahaabah asking.
That is generally the different opinions. Obviously, scholars from both sides go into more depth with regards to differences of opinion- we as laymen just know what is on the surface.
One very important thing to note though, and this is what the Ustaadh advised our class with... just because something is Mustahabb doesn't mean you should neglect it, rather one should strive to always do it. For example, salawaat after the final tashahhud- even if a person holds the opinion that it is Mustahabb, he should strive to do it all the time as although it is an obligatory act. Preserving recommended acts of this deen helps us preserve the obligatory acts more firmly.
WAllaahu A`lam
Wassalaamu `alaykum