First, what is the same

Same kernel,same .ipynb file

These are two interfaces onto identical machinery. A notebook written in one opens unchanged in the other, because the file format never differed.

The same kernels

IPython, IRkernel, and the rest. The execution side is shared machinery, not a difference.

The same file

One .ipynb, JSON, with cells and outputs. Portable between the two without conversion.

The same server

Both are front ends onto Jupyter Server. Authentication and remote access work identically.

What differs

One document,or a workspace

Classic Notebook shows you a notebook. JupyterLab shows you a desktop that happens to contain notebooks.

LayoutLab gives you tabs and split panes; classic gives one notebook per browser tab. That is the core of it.
What else is in the windowA file browser, terminals, text editors, CSV and image viewers — all in Lab, none in classic.
Debugging and inspectionA visual debugger and a variable inspector exist in Lab. In classic you print things and hope.
ExtensionsLab's extension system is the maintained one. Classic extensions are legacy and increasingly unsupported.
Which to use

Lab, unless you havea specific reason

The classic interface is in maintenance and Lab is where development happens. That settles most of it, but the exceptions are real.

Use Lab

Anything sustained: multiple files, a terminal beside the notebook, debugging, or a long-running project.

Use Notebook 7

You want the single-document simplicity, on modern foundations. The best of the classic argument.

Teaching

One document and nothing else on screen is genuinely easier for a beginner. This is a real reason, not nostalgia.

Notebook 7 muddied the comparison usefully: it is the classic interface rebuilt on Lab's foundations, so you get the simple single-document view with the modern extension system underneath. If the reason you preferred classic was the interface rather than the internals, that is the version to use.

On a rented machine

The kernel runswhere the data is

This is the part that matters more than the interface question, and it is the same for both.

on the machine

$ jupyter lab --ip 0.0.0.0 --no-browser

$ jupyter nbconvert --to script analysis.ipynb

On the machineServernext to the data and the GPU
In a tabYouno install, no SSH client
Idle timeWatchan open notebook still bills

Run the server on the machine holding the data and open it in a browser tab. Nothing is downloaded, the GPU is right there, and closing the tab does not stop the kernel. The thing to remember is that an idle notebook still holds a machine that is billing by the hour — stop the machine when you are finished, not when you next remember.

Restart and run allOut-of-order execution is the notebook's original sin. A result you cannot reproduce top to bottom is not a result.
Move code out earlyOnce a function is working, put it in a .py file and import it. Notebooks are a poor place for logic to live.
Strip outputs before committingEmbedded images make diffs unreadable and repositories large. nbstripout does it automatically.
On a machine

Jupyter Lab isalready built

Nothing here needs compiling. Pick a machine, and these two commands put it on there.

hub://jupyter

$ dxflow workflow create --identity jupyter hub://jupyter

$ dxflow workflow start jupyter

Read it, then run itPrepaid and by the hour, on a machine that is yours about a minute after you ask.