Principal and interest are weighted differently throughout the loan. You can generate the amortization schedule on most lenders' websites, and you will see the breakdown.
I only wanted to pay down part of my mortgage (as an insurance policy, leaving me the ability to recast in a moment's notice if I needed extra cash flow/ to reduce expenses). I decided to prepay as early into the loan as possible because I wanted my future payments to work harder towards the principal (effortless savings). There is a difference between paying down earlier during a mortgage vs paying down later into a mortgage; it makes a difference in how much you save because of how the scale is tilted.
Example of 2 different outcomes, using a lump sum paydown for simplicity:
1) $50,000 one time extra payment, 5 years into 30/3.5%/$250,000 loan = savings of $54,179.44, 7 years, 9 months early
2) $50,000 one time extra payment, 10 years into 30/3.5%/$250,000 loan = savings of $39,257.97/ 6 years, 7 months early
Difference in savings = about $15,000.
Same interest rate. Yes, the interest reamortizes during the loan (not the rate, but the real dollars), however the PITI payment does not because it is designed to be a fixed amount. But, as seen above, even though the amount is fixed, the allocation changes during the course of the loan and can be further manipulated by placing a proverbial thumb on the scale.