The nations of the UK are independent nations that share resources with each other, and sometimes a feeling of oneness and fraternity is seen among the good people of that land. Emotions are subjective. Would the rest of the UK have anything "real" to lose if Scotland exits the UK?
The Scottish government's 2013 white paper on independence - Scotland's Future - sets out a number of tangible assets that, the paper argues, Scotland would be entitled to a proportion of based on population share. For example, an independent Scotland would seek to take ownership of a share of the United Kingdom's overseas properties, e.g. embassies & consulates:
Scotland will be entitled to a fair share of the UK's extensive overseas properties (or a share of their value) allowing us to use existing premises for some overseas posts. For example, the Foreign Office owns or leases almost 5,000 properties overseas [source]. The estimated value of this estate is around £1.9 billion. Based on a population share (our actual share will be a matter for negotiation) Scotland would be entitled to around £150 million allowing us to establish ourselves quickly and for little initial cost in our priority countries.
Chapter 6 - International Relations and Defence
The paper also argues that Scotland would be entitled to £7.8 billion of the UK's existing defence assets - based on a population-share proportion of the 2007 valuation of ~£93 billion performed by the Ministry of Defence. It even splits out what particular assets Scotland would seek from the UK, including, amongst others, "one aviation unit operating six helicopters", "a Quick Reaction Alert (QRA) squadron incorporating a minimum of 12 Typhoon jets", "two frigates from the Royal Navy's current fleet" and so on.
Other assets mentioned in the report that an independent Scotland would seek to inherit include BBC Scotland, "assets that are not related directly to particular services, such as the UK’s public shareholdings in banks", "bespoke IT software", and so on.
Page 21 of the report does mention that Scotland may choose to offset its share of national debt against these assets.
There are also implications for the United Kingdom's nuclear deterrent - Trident. The white paper makes clear that an independent Scotland would no longer accept Trident being based at the ports of Faslane & Coulport:
The Scottish Government is committed to securing the complete withdrawal of Trident from an independent Scotland as quickly as can be both safely and responsibly achieved.
As the rest of the UK has no bases outside of Scotland suitable for Trident, this would present a significant problem for the Westminster government. Indeed, it has been proposed more recently that Scotland should fund its defence by leasing the bases out while a replacement is constructed - something which would almost certainly prove to be extremely expensive for the UK taxpayer. In a worst case scenario, the loss of the UK's nuclear deterrent would lead to a further reduction in its international standing. For example, the UK would become the only permanent member of the UN Security Council without nuclear capabilities.
From Wikipedia's article on It's Scotland's oil
Jim Sillars, former Deputy Leader of the Scottish National Party, said during the 2014 Scottish independence referendum that “BP, in an independent Scotland, will need to learn the meaning of nationalisation, in part or in whole, as it has in other countries who have not been as soft as we have forced to be. We will be the masters of the oil fields, not BP or any other of the majors.”
During the referendum, it was quite common to see pro-independence supporters using oil nationalization as a pro for leaving the UK. Whether or not independence could accomplish that is another matter entirely, but a lot of the economic rationale for leaving the UK was based on the assumption that they would get a lot of the oil rigs in the North Sea.
The UK would lose power. A bigger stronger country can field a larger army, wield a larger budget by collecting more taxes, and influence other countries on the world stage with threats like economic sanctions.
Arguably this should not be a consideration in modern Europe, since the citizens of the country itself don't benefit much from the country being more powerful, and we have international organisations to wield power collectively. Nevertheless, many world leaders seem to think this way, based on their hostile responses to independence movements and occasional attempts to expand.
Some nuclear weapons, potentially.
Scotland is home to various defense assets of the UK's military, e.g. nuclear subs.
If a part of a country seceedes, then the question of how to split up the military assets arises. Other than immobile assets, e.g. oilfields, military assets are quite mobile. While, you can't move a naval base can definitely can move out the ships. And a naval base without a navy is pretty worthless, unless it's in a region with expensive real estate.
When the Ukraine became independent of the Soviet Union, it inherited some of the nuclear weapons.
So, if Scotland became independent of the UK, it could inherit some of the nuclear assets stationed in Scotland. On the other hand, the international community is actively seeking to keep the number of countries at a minimum. Hence, the Ukraine disarmed itself nuclearily (is this a word? If not, I bestow my invention upon the world for free. You're welcome).
Thus, one could argue that the Remaining United Kingdom would retain its status as nuclear power, and Scotland will only gain independence non-nuclearly.
You could only find material evidence of the secession if you searched hard: it would be immaterial for families further south. It would be a loss of power for the UK government and for certain companies and banks.
For wealth, culture, the rest of the UK citizens would barely notice the difference. It is entirely a nationalistic and identity issue for the rest of Britain. Only Scotland would see major changes in the way of life because of new laws.