Settlement (aka indefinite leave to remain, ILR) means you retain your existing nationality, gain the right to live and work in the UK, but do not gain right of abode. To travel outside the UK you would still need to use your existing passport, and acquire visas as a national of your native country where required. You would still use the overseas channel at UK border control.
ILR can lapse if you stay away from the UK for longer than 2 years, meaning you would require a new visa. It is easier for ILR to be revoked than citizenship. Citizenship is (normally) for life, but using powers in the British Nationality Act, the UK Home Secretary can terminate the British citizenship of dual-nationality individuals if he/she believes their presence in the UK is 'not conducive to the public good', or if they have obtained their citizenship through fraud.
British citizenship confers some additional rights, such as:
- Ability to apply for a UK passport
- Reciprocal rights in EU countries (eg. health benefits, travel rights, working rights)
- The right to vote and to stand for public office.
Gaining British citizenship costs more and, for some countries, may affect your right to retain your original citizenship (aka dual nationality). The UK does not have restrictions on multiple citizenship, but other countries do. There may be tax implications of changing citizenship for nationals of some countries. Some employers may (illegally) favour job applications from British citizens.
Based on the description of your specific situation, you will probably be eligible to apply for either one of settlement or citizenship. Those pages give checklists and guidance.