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So far I did some homotopy equivalence problems, and they all absically boiled down to deformation retracts. For instance, I proved that a circle is the deformation retract of the moebius strip. Now I am aware if a space is a deformation retract of another then they are homotopy equivalent 1 but not vice a versa 2.

Could someone give some simple concrete examples which illuminates the difference? The example I saw of the infinite zig saw in the above post was a bit hard for me to understand.

I have seen these posts already : 1, 2, this very nice intuitive quora post

Clemens Bartholdy
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In some sense, all examples are deformation retracts. If $X\simeq Y$, there is a space $Z$ containing $X,Y$ and $Z$ deformation retracts onto $X$ and also onto $Y$ (see Hatcher, chapter zero).

But there are many examples of a homotopy equivalence where the two spaces don't naturally include one another. So, in particular, they are not examples of deformation retractions. One that I always like is, for $n>2$, where you have $n$ half circles attached at a common base and endpoint (formally, something like $\bigvee_n[0,1]$ with all the ending "$1$"s identified to a common point) and you know by some general nonsense it is homotopy equivalent with $\bigvee_{n-1}S^1$. A similar example would be that a pinched torus / doubly pinched sphere / croissant is homotopy equivalent with $S^2\vee S^1$, but there is no obvious way to embed one space into the other.

The "difference"? Well, if $Y$ does not embed into $X$ nor conversely, then $X\simeq Y$ cannot be written as a deformation retraction. For me, it's less that there is a difference and more that deformation retractions are just special cases where one space contains the other (and the homotopy has an extra property). Homotopy equivalences, I'll grant, are much harder to visualise as "wiggling space around" when they're not deformation retractions, but ... this is just the general, useful notion. Maybe the most important point is that the maps $X\to Y\to X$ need not be "faithful", injective, surjective, whatever; in a deformation retraction there is a clean inclusion to focus on, but in general you could have quite weird maps which witness the equivalence.

FShrike
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An answer here depends on exactly what you mean by "boils down to deformation retracts".

If you're asking for two homotopy-equivalent spaces such that neither is a deformation retract of the other, then the most straightforward example I can think of would be the letters H and X, written in $1$-dimensional strokes.

If, on the other hand, you don't even want the two spaces to be deformation retracts of a third space, then there are no examples! It turns out that any two homotopy-equivalent spaces are both deformation retracts of the mapping cylinder of a homotopy equivalence between them (the proof is involved enough that I won't reproduce it here).

So in some sense, homotopy equivalence really does boil down to deformation retracts.

MartianInvader
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