I can't tell how close these spaces are to each other. Are they separated by any walls at all - even partial walls with a large opening?

Very close

I have a feeling you are trying to keep a traditional formal entertaining area, but want a more comfortable spot for everyday usage.

They are not working style, size, and placement wise, so believe editing/removing on your part will be needed one way or the other. If interested, seeing a picture of the entire room will help layout wise; combining comfortable/traditional is possible with sectional, if willing to make big changes and whatever is the best option for your needs. How you accessorize can also bump it up a bit also.

Yes, sorry, you're right, they are too different to be right next to each other. Pick a direction and proceed accordingly as Maureen says. Is part of the space meant to be a dining area? Maybe you should just dispose of the formal seating and buy a table? Without seeing the layout of adjacent rooms I'm not sure if this is a good idea or not.

You can’t make those work together. Also, the sectional looks oversized for the space. I’d remove it and stick with a classic style for that room.

I am guessing that the formal furniture is from your family and you are keeping it out of sentiment. Decide how you really want to use the room and go from there. Maybe keep one formal piece and replace the rest for comfortable seating.

Unfortunately, it's tough to make those work together unless you were willing to reupholster the French chairs with a modern fabric and maybe even lacquer the frames. In your case, I'd probably do a black lacquer for the frames. Here's an example in white:

It would give your space a more eclectic look. Once you pick the fabric, you can then change out the rug, too, to something more modern.

Replace the occasional tables, too, to something more contemporary and light-feeling.

I need to see the whole space of each room and also the use of the2 spaces. I do agree the sectional is too big for that space I do not think you can really make these 2 spaces work together . I do love the chairs posted above but IMO they belong in a really contemporary space and neitehr of your spaces are the right mix for the chairs.

You could just change out some, or all, of the pillows on the sofa to a more casual fabric pillow that contains the color of sectional. I am not exactly sure what color the sectional is - but it looks like a blue on my screen.

And I would put the wood coffee table in another room, if possible - and bring in another coffee table; and I like the idea of something like these nesting tables.

Then, on the sectional - you could add a throw, along these lines.

And/or some floral/damask pillows on the sectional - in the general pattern/style of your rug(s) - and/or you could use a couple of pillows on the sectional along these lines.

Please post more photos so that we can see the entire room (all 4 walls and/or entryways).