Have you tried the "curve project" function to make a curve of your shape to be extruded on the outside of the tube, then use the "split" function to cut the tube with this curve, and finally "remove body".
Just a silly question: is it the same function (or sketch) applied in the 2 softwares?
i.e. do you do a tube/tube removal like what you drew on SW?
Because if you take your laser head fixed, your tube rotates on itself to be cut, you don't get the shape of SW, but a propeller. Well I may be wrong...
edit: so it comes back to the first answer of PL. do it in 3 functions.
You can't see well in your pictures, but I have the impression that you have converted to sheet metal, but it is not a sheet metal part in the sense that it cannot unfold: your basic shape before material removal remains a tube, and not a rolled sheet.
When a part is laser cut, the head remains fixed and the tube rotates, so the laser is always normal to the cut surface. This is what I would like to reproduce on solidworks, to be able to send the file to the laser. Today when I send it it it cuts following the outer edge of my drawing so the cut piece does not correspond at all to the desired one.
I work in 3D laser cutting so my laser is able to follow the external curve by tilting its head: up to 45°, you can't do everything anyway:)
To help you, you should keep the inside of the tube as a reference in sheet metal construction
and perform normal material removal.
See attached file made on Solidworks 2015.
I created a model of the sheet metal tube with a flat face Lg 0.05mm (which my cutting software eliminates because it is too small) Then I create the normal material removal on the surface.