What is the difference between spectrum sequences and spectral sequences? Are they considered to be the same?
I know that the spectrum sequence of a real number $\alpha$ is the sequence that has $[n\alpha]$ as its $n^{th}$ term.
Can spectral sequences be defined the same way?
They are pretty different. A spectral sequence is a computational tool to approximate (co)homology groups in homological algebra and algebraic topology. The sequence part in 'spectral sequences' comes from the fact that you take successive approximations to (hopefully) converge to the correct group (so a sequence of approximations in that sense).
If you have a basic understanding of abstract algebra and a lot of patience, you can follow the construction of a spectral sequence via exact couples, although it might be unenlightening without some background or motivation from homological algebra or algebraic topology.