What are the fundamental differences between a PMM and a senior PMM? What are some clear indicators that someone is ready to take on a senior PMM role and the responsibilities that come with it?

This is a hard question because I've found that even the nature of product marketing and the title "PMM" means different things at different companies. Generally, however, I'd look at the framework for seniority against the following dimensions: 

• Product coverage: Early in your career you may manage a few products, with "few" defined relative to your company's portfolio. As seniority grows, you'll stop owning products but start owning categories of products. The idea here is that at a more senior level you'll be trusted to not only tell the story for the products but also uplevel them into meaningful value by audience, by use case, or by some other dimension. 
• Scope of responsibility: Often your job starts with launch (the bread and butter of product marketing!). As you get more senior it will grow and can take on dimensions of sales enablement, campaign design, thought leadership, analyst relations, PR, public evangelism and more. 
• Stakeholders: Typically the first level of PMM will interact closely with a few PMs and probably some members of associated teams across marketing and potentially customer engineers / solutions consultants. Basically folks who have an intimate interest in your products alone. As you get more senior, your relationships will both elevate in level (ex. from PMs to PM Directors to VP PM) as well as a diversity of teams (ex. from Marketing + PM to add on Sales, Sales Enablement, execs, Legal, Analyst Relations, even outside agencies and influencers). A senior PMM deals with decisions that are more complex with larger consequences.
• Visibility: This means internal visibility - how well people are observing your work - as well as external visibility - how well people know you. More junior PMMs often work on projects that are critical but less visible, meaning that you have often more room for leniency if something doesn't go as planned. This is a good thing. As you move up to a senior PMM, the projects you own often have executive interest and that will increase the complexity of your decision-making process as well as the precision with which you need to execute. Senior PMMs also frequently meet top customers and speak publicly at events. 

The big thing is to keep in mind what you are learning. As long as you are learning and stretching, and demonstrating that to your leadership and those who influence the advancement process at your company, you will grow into a senior PMM or a more senior PMM. I look for evidence of working at the next level, and for a sustained period of time, as I've worked with team members to grow in their careers.

For me, it was primarily just based on experience, and having done a number of launches, GTM planning, messaging, buyer/market analysis, etc - that I know what to look out for, and don't need a lot of hand holding. You can read about how to do our work, but until you are in the trenches and learn what really works, what doesn't, and what to look out for, it's hard to understand what really goes into it. As a PMM I can hit the ground running in nearly any company, because I know what's expected, I know what has the highest impact the quickest, and I need very little guidance/direction. I've also learned what "red flags" to look for, having gone through the growing pains, and how to spot/correct them or avoid them all together.

My work is probably less scrutinized than a PMM who may be just getting started (or thinking back to when I first started). I'm not learning as much about PMM itself, as much as I'm perfecting my craft and skills in the field. I'm still learning of course, but it's how to go deeper, and how to scale the work that I do.  

I'm not sure it's really any MORE responsibility, in terms of tactically doing different things. I don't do anything more than a non-senior PMM does, I just may be more efficient at them and need very little direction on a path forward. I also know what things to focus on, track, measure, etc - those I think are really tough for people to grasp early on. If you have an eye on how the work you're doing is influencing the product's success throughout the year, then you're ready. 

• As with many roles, a PMM already demonstrating they are working above their level is potentially ready to move up. Here are some further thoughts to determine whether someone is suitable to take on a senior role.
  
  Safe pair of hands - consistently delivers on goals and achieves significant milestones in time or ahead of time but not at the expense of others.
  Performance - is performing above average and shows high potential to continue to improve.
  Expands scope - grasps new opportunities offered. Finds solutions to problems.
  Good partner - considers others' points of view, treats others with respect, and is inclusive.
  Contributor - contributes to meetings and team projects but doesn't dominate every discussion or project.

There are already some good answers on this topic. However, I would like to add that beyond scope, impact, and stakeholder influence, the main shift between PMM and Snr PMM is autonomy.

What does this mean? It means seeking out areas for improvement and problems that need to be solved without managers providing a step-by-step list of things to execute on. This is proactive problem solving without manager intervention.

Another way to think about it is being proficient in the hard skills of the PMM who can break down work into deliverables and execute as a dependable IC. This is the foundation to then becoming a team manager for PMM where you have the ability to coach teams members on those skills.