When the number of discount factors that define a curve is larger than the number of market instruments, the interpolation limits the shape of the curve, and that fixes the discount factors. The quantity you interpolate is the discount factor, its logarithm, the zero rate, or the forward rate, and the quantity you choose sets the shape of the forward rate between the points.
Consider a hypothetical example in which the 1-year discount factor is 0.970446 and the 2-year is 0.923116. Linear interpolation of the log discount factor gives about 0.946485 at 1.5 years. Linear interpolation of the discount factor itself gives about 0.946781. The same two points give different values in between, depending on the method.
In a simple form where the values at the points are fixed first, interpolation leaves the point values found from market rates unchanged and determines only the values between them. In an actual curve construction, the parameters of the interpolation formula are solved to match the market rates, so changing the interpolation can also move the values at the points. The differences between methods appear in the present value of cash flows at maturities with no market rate, and in the forward rates.