Hello!
Firstly, I would like to start by saying that I can only speaking confidently about the experience at LSE, my university, whilst my insights about Cambridge's economics degree are solely based on my research and the experience of some of my friends. Okay so let me get into some of the differences. Number 1, LSE econ is designed to be very mathematical, no matter your outside optional modules. Especially the second and third-year compulsory modules I would say can be very mathematical and rigorous in nature. For example, if you choose to do Microeconomics 2 EC202 or Principles of Econometrics EC221 (which is a very prestigious and a gold standard course, which provides a great foundation for graduate studies) are a lot more mathematical than the micro and metrics courses taught at Cambridge. That said, the alternatives, so EC201 and EC220, are still very rigorous and mathematical but are not much different from those taught at Cambridge. Third-year courses, given what I've heard from older students, of course, are very very challenging and you can think of the first two years, as prep for the third year, where you put all the mathematical and somewhat basic ideas into the test, through applications to the real world of course. Secondly, something which is quite different and known, is that LSE economics is designed to be taught in a way, that fully prepares you for graduate studies, which isn't really the case at Cambridge. That means that, if you know that might be something that you're interested in, LSE provides a great opportunity to prepare for that, for example through EC221 as I said, or MA203 a real analysis course taught at LSE. I do not think opportunities such as these, so to take courses that prepare you for grad study or simply take courses that provide you with a challenging economics education at undergrad, are available as much at Cambridge. Three, at LSE you can choose to steer your degree in the way you want really, so it's a lot more flexible. Yes, some of the courses are a lot more mathematical and challenging than Cambridge's, but through your outside options, you're really given the opportunity to explore modules from other departments, such as law, politics, anthropology, international relations, and so on. I know people who do out compulsory courses and are doing those that are more on the qualitative side and have chosen to completely stay away from econ for their outside options. They still graduate with a degree in econ
Not that much flexibility at Cambridge. Although I seem to be very biased here haha, no one can argue about the standards of education and rigor at Cambridge, it's all about what you want to get out of your econ degree.
In terms of student satisfaction, it's mainly really what you make out of it. Of course, you do want to have a good think about whether the differences in the place, for example, whether you prefer the diverse, fast and busy city or not, because these will contribute greatly to your satisfaction at university. But in terms of societies and making friends, they're both places that have some of the world's best students in their chosen fields, so of course, you'll find those that are more on the introvert or anti-social side but you'll also find those that are eager to make friends and have strong relationships with people. The point is if you go to either university and decide to put yourself out there and make friends, then I guarantee you'll find the right people through societies or halls for example.
Onto the new economics degree structure. Firstly, there isn't any change to the third year, which is great because this is mainly where you bring everything you were taught at the first two years and choose where and in which field you'd like to apply that knowledge. So it's still quite flexible. Generally, I do think it's a change for the better and you guys will probably find this to be a better educational experience, especially for the first year. For one, you're all now being taught the same introductory economics courses, unlike in previous years where we had a course for those who studied A-level econ and another for those who didn't. This is great because no one really understood how two different courses are meant to take everywhere to the same start point for the first year, and it didn't. Something like game theory, which is explored in quite some depth in Micro during the second year, was covered in the economics module for those with prior econ knowledge. While the other portion of students is introduced to game theory for the first time in the second year. You now do not have to go through this. One other change that I notice is that the first term in the first year seems to be a preparation for the second term, which is again, something great. Easing into the courses is something that you'll really appreciate if you join LSE, given that you have some time to adjust with fairly simple ideas. In the first year, we were never taught econometrics, and we did micro in the first term and macro in the second. This means that you're taught stats in the first term and then taught how to apply that knowledge to an economics problem (econometrics.) This also means that, while we previously did micro first, got it over with, and kind of just forgot about it until the start of the second year, you do not go through that anymore as well. Instead, doing these two side by side, having studied an intro to econ in the first term, will help you see how these two come together to really shape society. it will definitely also set the tone for the courses in the second year because you do study these simultaneously as well
Finally for the second year, it seems they have chosen to remove the option of which micro or metrics module to study and this means that they'll probably find a better balance between sustaining the mathematical and rigorous nature of the more challenging options, while still making it a little more qualitative and less complex. I am a little doubtful about this change because it no longer allows students to, as I said challenge themselves or just simply get by. So, to accommodate for both, it could be seen as a little simple for some but a little complex for others.
Hope this helps