If the UK leaves the Single Market but stays in the Customs Union, what will be the difference, if any, to what we have now?
The European single market guarantee the four freedoms: free movement of goods, capital, services, and labour. The customs union is merely about free movement of goods. So compared to the customs union, the single market notably also includes the famous "financial passport" for banks and financial companies, freedom of movement and establishment for citizens, freedom to provide services (including digital services). The customs union already requires the country to let the EU negotiate all trade deals (for obvious reasons).
The only countries which are in the customs union but not the single market are Turkey and some micro-States (Monaco, Andorra, San Marino). Within the single market, you also have to distinguish the EFTA (which participates in the single market with some minor exceptions), countries that are in at some stage of joining the EU and have entered a SSA with the EU, and countries that have signed a DCFTA treaty with the EU.
The respective Wikipedia pages are very detailed, with lots of sources, and provide good starting points:
Leaving the single market (SM) but staying in the customs union (CU) makes differences as follows:
Leaving the SM would make the UK:
- Able to apply tariffs and taxes to EU countries, and vice-versa.
- Able to prevent imports of goods from EU countries,and vice-versa
- Able to prevent immigration of Poles, etc, and vice-versa.
- Able to create non-tariff barriers by, for example, abandoning or de-fanging Health and Safety regulations, and vice-versa.
- Able to prevent EU-based financial services operations (Goodbye, Santander?), and vice-versa.
Staying in the Customs Union would make the UK:
- Unable to negotiate its own global trade deals.
- Liable to pay the EU for the cost of membership
- Subject to the decisions of the European Court of Justice (ECJ)
- Safe in regard to Irish border problems
- Unprotected in regard to farming and fishing