Different magnitude values when computing f-domain function by hand and with fft matlab command.

Hello everyone, I have the following function in time-domain and its correspondent f-domain transform:

With t1 as [0s,8s) and f1 as [-50hz,50hz)

When I plot, I get the following:

Now, I tried using fft command to the time-domain function and see what I got, even if I choose the same t1 and f1 vectors. For my surprise, we have differences when plotting:

Why's that? Is it something about fft command?

Here's how I'm placing the formulas in case you think there's a mistake in there:

s1(t)

s1=(t1-2).*exp(-t1).*heaviside(t1-2);

S1(f)

S1f=(exp(-2*(1+(2*pi*f1*1i)))./((1+(2*pi*f1*1i)).^2));

Can you help me find a reason why is this happening?

Thanks.