Why Pakistan and India have different claims over Kashmir?

What is the Kashmir issue between India and Pakistan?

Fair-Warning: This will take a lot of time.

Let's tackle this in 2 ways: First we look at current-claims, objectives. Then we look at it on a chronological order.

Current-objective: Pakistan wants Kashmir to be "Azad" - if you do some quick research on PoK, you'll know what this "Azadi" really means. India, for the most part, just wants the violence to end - and they're ready to seal-the-deal with current borders as the permanent boundary. Like Indians elsewhere, the current natives living in Jammu also want progress & development. One can't confirm the goals or objectives of the current entities living in Kashmir, though. I'll explain why in part-2 of response.

Sources: What "Azadi" means for the PoK

What India wants today?
1. India Willing To Turn LoC Into Formal Border
2. Hurriyat attacks Omar Abdullah for proposing to convert LoC into border

Historical perspective:

a. Jinnah's calculation regarding Hyderabad-Nizam (current day Andhra & Telengana) and Junagadh
It is much easier to just read this article: "Pakistan's lies on Junagadh and Hyderabad"
1. Junagadh & Hyderabad were hindu majority regions which were miles away from pakistan. 
2. Its muslim rulers decided they didn't want to join India. The nawab of hyderabad wanted his own separate nation; while the ruler of junagadh wanted to join pakistan. 
3. When these things were officially announced all sorts of horrible things happened. 
Junagadh: Huge number of people from Junagadh ran away - fearing it will become part of pakistan and the kafirs will be slaughtered [past-experience has taught them well]. Eventually, because majority of the people fled the kingdom and consequently almost everything came to a stand-still:  the ruler of Junagadh himself fled - taking his kingdom's valuables, money with him - to pakistan.
Hyderabad: Nawab solicited assistance of pakistan - and he gave them huge amount of money (government bonds valued at 2 million at 1947) and set-up his PRO in Karachi. However, he confirmed to the British that he will not accede to either pakistan or india; and would be a separate, independent region. He passed ordinances preventing trade-movement and Mountbatten opposed these ordinances. To bolster his position of being an indepdent nation, the nawab contacted POTUS for intervention. No help came through. Then, he filed a case with the UN. While doing these steps, he also had his private militia control the people [remember the people were hindu majority and they had supported gandhi for freedom - not to be part of nawab's kingdom]. Due to the violence, several people started running away to Madras province; this inevitably caused Indian govt to redirect army in Madras province. After widespread violence and excessive loss of human lives, with no alternative choice, Nehru unwillingly issued orders to Indian army. The army entered hyderabad - the private militias surrendered. The Nawab withdrew his case from the UN and acceded to India.

Jinnah calculated that Nehru would forcibly invade Junagadh & Hyderabad - and this could be used as a pretext to invade Kashmir. Unfortunately, in both cases, Nehru under strict supervision of Mountbatten - gave complete freedom to the kings to work things out with Jinnah (& pakistan). After waiting many months and after the kings breached the "stand-still agreement" clauses, and after several warnings & meetings, Nehru was forced to use Indian army. Since Mountbatten was supervising all this: the information was being sent to UK, USA, UN and other nations. Thus, when Indian army took control none of these powers raised even a finger.

Thus, Jinnah's calculation had failed.

b. Post 1948-war - and the infamous UN resolution [blunder by nehru, as per some] [UN Security Council Resolution 47, Kashmir]
The UN resolution calls for following steps to occur:
step-1 - Pakistan must remove all its armed forces and invaders & occupiers from PoK [Point A - 1 - a) - on page-4 - the very first point in the resolution]
step-2 - India must integrate the people of PoK, along with J&K and arrnge for a plebiscite
step-3 - UN Peacekeeping forces will be involved along with UN oversight/supervision for both steps 1 & 2.

EDIT: Update as per "Utpal Baruah" [comment dated 29 Sep 2015]

The UN resolution says that Pakistan will withdraw ALL it forces from Kashmir and India will maintain a MINIMUM amount of forces to ensure smooth conduct of plebiscite.
So, effectively it means handing over entire Kashmir, including the area occupied by Pakistan currently to India first and then Pakistan can demand a plebiscite.

Source: United Nations Official Document

The UN & the world is still waiting for step-1 to even begin. Indian perspective: In the past six decades, the demographics in PoK has been forcefully changed. So, it no longer makes sense to hold a plebiscite.

c. Post 1965 & 1971 & 1999 wars: After every single war - the agreement was to disregard the UN resolution of 1948 and instead solve the issue through bilateral talks [no third-parties, no UN]. And every single time, the Pakistanis have signed the agreements. However, most pakistanis don't know these agreements were signed - because as per most pakistani school text books, Pakistan has won all these wars [may be except 1971] - so it makes no logical sense to see why the winners will want to leave the UN out of this. Hence, most pakistani public still talk about the 1948 resolution [without realizing that the very first step requires pakistan to withdraw its invaders, occupiers, armed forces out of the land - doing so would be a tacit-acceptance that in deed Pakistan was the aggressor in 1948 - which contradicts its official position] [If possible, please listen to the entire video: if not at least the from the 10:00 minute mark till the 30:00 minute mark: 

Conclusion: India is a status-quo state in regards to geographical boundaries. It doesn't want to expand - it doesn't want to usurp Tibet or Nepal or Bhutan or Burma/Myanmar or Maldives or any smaller nations near-by. Pakistan is a greedy-state [not my choice of words: this is sourced from Dr. C. Christine Fair from GWU George Washington Uni who has lived in Pakistan and worked at LUMS,Lahore] - it wants to change the current boundaries. This is the political-cause behind the violence.

Personal-Opinion: Even if Kashmir & even Afghanistan both issues are solved - and they're solved exactly how the Pak-Army wants it [say entire Kashmir is given on a silver platter to Pakistan; and the Taliban were to rule Afghanistan, as before] - the Pak-Army will still consider India a threat. The simple fact is: if you don't have an external threat, you can't justify the amount of resources used by the army. You can observe this in G.W. Bush's america too: Iraq was a threat [till now (Sep 2015) no WMDs were found], Afghanistan was a threat [whereas not a single Sep/11 attacker was of afghan nationality]. You need a bogeyman - in order for the military-industrial-complex to successfully loot money from government into private defense manufacturers. This is the base-line fact of the matter.

Apologies: If I've misrepresented anything or offended any Pakistani (or anyone else, for that matter), please do accept my apologies. I don't believe that the Pak-civilians have any major role in any of the above events. I don't think civilians of any nation deserve to die; and I don't think military personnel deserve to be killed by fidayeen attacks [both in Pakistan & India - and elsewhere in the world]. Wars must be fought based on Yuddh-shastra [or, Rules of Engagement] - and civilians must not be impacted due to wars.