What do you consider "normal" filming?
I've filmed a lot in theatres with audiences, usually in performances that are primarily for the audience, and the video recording is a secondary consideration. Usually the camera in this scenario is back far from the stage (behind the audience, so you're not in anyone's way), and the camera positions are fixed. You're capturing the performance live, and don't know exactly when/where the performers will be, so framing is usually paying attention and a bit of guesswork. The focus is usually fixed in the right general area, which is fine for wide shots from up the back of the theatre. If anyone on stage has a mic, usually there's a clean audio feed to the camera of that, as well as any music (although you can also re-dub anything from a pre-recorded source later). Other than that, there is also usually an auditorium mic for the camera, to pick up any other audio (non-mic voice/singing, any other live sounds). Lighting is also usually for the "live" show, and from a camera point of view you're just having to adjust on the fly, and cope with whatever you're given.
There are various pros and cons to filming without the audience.
Pros: From a visual perspective, with no audience you can bring the camera much closer, which will give you a better (sharper/detailed) image. Your perspective is usually more in line with the stage (level) than what you can get from further back. You can also start choosing from more camera angles - if the material calls for it, and depending on how much editing you want to do. During COVID, I saw a few people film more in the style of TV production - you can block out exactly where everyone is going to be, and get in close (even on stage) with DSLR setups, different lenses, etc. It's a whole different look than what I normally do with video cameras from the back of a theatre. You can also control the lighting to be camera friendly, and make sure lights/colours that are unfriendly to cameras aren't in use. Being up closer means better sound recording also, and no audience sounds getting in the way.
Cons: DSLR filming is a different skillset to using a Video Camera. I'd be comfortable guiding anyone on video camera setup for a live show, if you handed me a DSLR to film a non-live setup, I feel it would be like starting with almost no experience - I'd need some guidance myself! Remember to consider that the two scenarios are different, so if you train someone only in filming up close with DSLR setups, they can't directly just swap over to filming a live show with a video camera, and vice versa. Secondary to that is the lack of audience - if you film without any audience sound (applause, laughter, reaction, etc), your performance will sound dead. If you break items up into scenes, and then re-dub your audio track later, it won't sound like it was recorded in a theatre "live". It will depend on whether you want it to still look/sound like a production that was recorded "live", or if you're happy for it to look/sound like it was performed for camera only. I filmed some dance numbers without an audience - the director decided that with no audience they could therefore give stage direction during the items, because I could just use the music audio track only, and didn't need to use the live mic sound - to me the final product looks like it was filmed in a theatre, but doesn't sound like it - it sounds like a studio recorded track, not a live theatre performance recorded track - any sound of any footwork is lost, etc.
Hopefully that gives you enough to consider. As always there is no "correct" way to do this, it's largely a bunch of trade offs to consider, which affect what the finished product looks and sounds like versus what you want to achieve.