I'm coming to the end of HMRC's TPDP programme so I can clear up a few misapprehensions.
First, AAT accredit the first two years of the programme but you do not study for the AAT award. After two years you receive a Diploma in Taxation accredited by the AAT because there had to be some external accreditation of the early years if HMRC wanted its trainees to go on to receive a degree. It could have been the ATT, City & Guilds or any number of other external bodies but they have chosen AAT. The first two years you study modules in bookkeeping, self assessment for individuals, business economics, VAT, intro to corporation tax and employment income alongside HMRC internal stuff like penalties and the rules on tax enquiries. In years 3 and 4 you go on to study things at Level 5 & 6 of the national qualification framework (final two years of a BA degree). The subjects are on business profits and tax cases, advanced corporation tax and capital gains for individuals and companies. The final two years are accredited by MMU and you get a degree in Taxation. In the past the HMRC graduate programme has offered a significant number of exemptions from the CTA qualification and I understand the new TPDP programme will offer something similar (although if you want CTA you fund it yourself and study in your own time).
On completion you become a grade 7 with a starting salary around £47-50k depending on London or outside London. From here you can work in tax investigations for large businesses or SMEs, become a technical specialist in a small area of tax, become a team manager, work in a policy role and loads of other opportunities depending on what jobs need filling at the time. Grade 7 is also a Civil Service wide grade and you could move to a non-tax job at Grade 7 if you wanted to. After grade 7 is grade 6 and then the Senior Civil Service and there are opportunities appearing regularly at the moment for those grades.
Salary wise it is a pretty good training salary, and compared to friends in Big4 firms the early career 'qualified' salary is good as well. But as you get more experienced the private sector does have much better salaries, along with healthcare and gyms etc, etc. But HMRC does have a good pension (even after the government has fiddled with it), is very family friendly and flexible and there are some very well respected tax brains in HMRC that could earn more in the private sector but choose not to because they balance the benefits against a better salary and stay put. It's a personal choice.
In the office during the 4 years of training the work starts off with small owner managed businesses and you are working with someone else or on your own in a civil tax investigation. As the stuff you are learning gets harder and more technical so your manager will provide you with work that matches it. In the last stages you will be working with large businesses on technical matters, often agreeing tax treatment of transactions in 'real time' (rather than waiting for a tax return to come in and then query the tax treatment) and negotiating with the in-house tax teams of multi-national groups or big 4 staff.