Which differences are there between Math PhD programs in the US and in the UK regarding admission, funding, and reputation? I'm an American citizen and a Junior at a US institution and I've recently been compiling a list of graduate schools to apply to. I'm mostly interested in differential geometry and geometric analysis. In addition to domestic institutions I have been looking at institutions in the UK such as the London School of Geometry and Number Theory, Cambridge, Oxford, etc... 

Are there any fundamental differences in the way UK PhD programs admit and train their students? As an international student would I also be able to obtain funding for my education the same way I would be able to in the United States? Finally, how are UK PhD's looked upon in the American Academic job market? I imagine there shouldn't be much difference but there could be some politics I'm overlooking. Thank you.
I am a first-year PhD student at Oxford, having done some graduate work (and my undergrad) in the US. My answer might be a bit institution specific as Ruadhai Dervan suggested above, but it appears there are some trends across the UK that are quite different from in the US.

"Are there any fundamental differences in the way UK PhD programs admit and train their students?" - Most students seem to enter a PhD program in the UK with a fairly specific field of study in mind. For my applications to Manchester, Warwick, and Oxford I had to write a research proposal for a fairly specific topic for each application, as well as have an idea of who might supervise this research. Many students deviate substantially from these ideas, but almost all students here work in a very similar area to what they applied for. In contrast, I know many US students who started out doing pure mathematics preliminary exams and found themselves finishing a thesis in a very different area, such as Numerical Analysis or Mathematical Biology. These things happen much more infrequently in the UK, in part because the programs are much shorter. The Oxford PhD is designed as a 3 year course, though many students take 4 or more years in practice. Some institutions are starting Doctoral Training Centres where a first year of training and project mentoring occurs. This is similar in flavour to US institutions having preliminary or qualifying exams, but these are often much more directed than in the US. I have not heard of any UK program where anything like preliminary or qualifying exams takes place, so that appears to be a US component of graduate school not present here. I would suggest looking at specific institution overviews of their PhD courses to get an idea of how they compare to US PhD programs.

Another important difference is the emphasis here on research as opposed to teaching. Most students in my program do some teaching, but it is frequently less than three hours a week. In comparison, my MS program in the States had me teaching 10+ hours a week, often with 10 or so hours of prep and grading. I also began doing research the week I arrived, whereas a US student may spend some years deciding on a specific topic to study. PhD students here also spend less time attending classes, and coursework requirements seem to be incredibly minimal compared to most programs in the US.

"As an international student would I also be able to obtain funding for my education the same way I would be able to in the United States?" - Probably not in general. Most US students in my experience are funded as Teaching Assistants, and they spend a lot of time teaching. Students are paid for their teaching services here, but the quantity is so much less that the funding from teaching is nowhere near enough to cover tuition and living expenses for most students. Exceptions probably exist, but most students here seem to be funded from grants their supervisors have, grants or scholarships they have gotten (e.g. Fullbright, Marshall scholarships), or similar things. You can apply for loans subsidized by the US government as well, just as you would for a domestic program. Many students are now funded through the Doctoral Training Centres, but at Oxford the funding for these is almost entirely for UK/EU students.

Your third question is a bit difficult for me to answer without any experience, so I will refrain from conjecturing. Most people have suggested to me that other factors, such as quantity and quality of publications, conferences and connections made, etc, are more important than where you obtained your PhD.

I would suggest finding a copy of Steven Krantz' book4, and read through it as soon as you can. It is directed almost entirely at students applying for or entering US programs, and is useful just to get an idea of what the whole enterprise of a PhD is about.