In 2015/16, on a £60,000 profit as a sole-trader with no reliefs other than the personal allowance of £10,600, you would have paid income tax of £13,403.00, Class 4 NIC of £3,441.55 and class 2 NIC of £145.60, a total of £16,990.15.
If you made a £60,000 profit in a limited company, you would pay corporation tax of £12,000 (20%). Personally, you would only have tax to pay if you drew a salary and/or dividends. Many sole directors draw a salary below the point at which they and the company pay Class 1 employee (12%) and employer (13.8%) NICs but at a level at which they are given credit towards their state pension. Any additional income can be taken in the form of dividends which are not subject to NICs and up to 5 April 2016 at least, were treated as basic rate tax paid. So, if you drew a salary of £8,000, you would pay no tax or NIC on that. If you then drew £30,946.50 in dividends, they would gross up to £34,385.00 with a tax credit of £3,438.50. As your total income would be £42,385, you would be a basic rate taxpayer and have no tax to pay personally.
In the company, the salary would be a tax deductible expense saving the company 20% in corporation tax. Dividends are not a tax deductible expense, just a distribution of post corporation tax profit. The rate of corporation tax is due to fall to 19% from 1 April 2017 and to 18% from 1 April 2020.
From 6 April 2016, the way dividends are taxed has changed. The sum drawn as a dividend is treated as the gross income. There is no longer a tax credit. If you draw a salary of £8,000 and dividends of £35,000 in 2016/17, you will pay tax of £2,025 on the dividends (£35,000 - £3,000 personal allowance balance - £5,000 @ 0% = £27,000 @ 7.5%). If your income was over £43,000, the dividends in the higher band would be charged to tax at 32.5%. You can experiment with figures using the calculator HERE.
As far as your car is concerned, you should be putting through a claim for capital allowances or a business mileage allowance. The CAs or the BMA will reduce your profit for tax purposes. You could claim a BMA through a limited company. The company would save corporation tax at 20% and you the cash would be tax free in your hands. You probably would want to avoid having a company car as that would give rise to a taxable car benefit and a fuel benefit if all your fuel costs were put through the company.
I hope this helps but let me know if you have any further questions.