Both web consoles are basically lag free and update immediately (I tried on a 1gig wired network and using wifi), but the refresh rate seems a bit faster on the Rigol. The main issue for me on the SDS1104xe is that the Siglent web console doesn't render properly in Chrome on linux, all you initially see is the waveform part of the display which doesn't accept clicks, so it makes it look like it's not working at all. What is actually going on is that all the buttons/controlled are rendered outside of the area of what you initially see, so you have to scroll horizontally to the right to see any controls. I attached a screenshot showing how wide the window has to be for siglent so you can actually interact with the scope controls. A side effect of the scope not having a touchscreen- the web control replicates the hardware knobs and buttons to the right of the display, so you need a really wide window. The other screenshot shows you have to click on the little arrow button at the top right of the window for the scope controls to be visible, the fullscreen button doesn't work because it only puts the waveform part of the display fullscreen.
Screen captures were easier for me on the Rigol, because of the rendering bug on the siglent. But Rigol has its own bug involving screen recording- if you turn on screen recording while you're using the web control to drive the scope, the web control window stops updating. Also Rigol doesn't seem to have a way to save waveforms from the web UI, only screenshots and videos. If you save a waveform on the Rigol directly, however, it at least lets you choose the file format as .bin, .csv, or .wfm and you can choose whether the data source is the screen or the capture memory. A more basic issue as far as connecting to a network in the first place is that Siglent has native support for USB wifi dongles, but the Rigol does not. You can get wifi working on the Rigol (using the steps in that blog post I linked) but the scope UI will not give you any indication that it's enabled/working.