To 'co-ordinate' means to bring two or more aspects of something together, and to make sure that they work well in relationship with each other. If you co-ordinate your outfit, it might mean that all the colours of your clothes match and look good together. In a working context, if you co-ordinate a project, it means that it is your responsibility to ensure everything happens at the same time and that the different aspects of it fit together as they should. I'd agree with Gary that 'co-work' is not a usual verb, at least not in British or Aus/NZ English. It is clear what it means, though - it just means work together. I suspect that it isn't as close as 'collaborate', however. I have frequently come across the term 'coworker' in American English, simply meaning 'colleague', so I would guess - as Gary says - that it probably refers to people who work in the same office or company as you.

You would never use "co-work", but you might use "co-worker" which means someone who works for the same company as you do. The "Co" is the same as in co-ordinate or co-operate and implies something done together or in a group. Americans would leave the "-" out sometimes and just write coordinate or cooperate. Co-ordinate means to manage a group. For example, if you have a group project you would assign a project co-ordinator to manage the group.

The difference to me is that coordinate is English and co-work isn't! Although I have just looked cowork up, and it seems to be used a little. From what I can see 'coworking' just means 'sharing a office'. Perhaps it's just a marketing term invented by people that want to rent out shared office space?