First of all, what Doctrine are you talking about, 1 or 2 ?
There is a huge difference. The only thing that the two have in common is that they are both full-fledged ORM-s. Otherwise there really isn't any connection between the two.
Doctrine 1 is based on ActiveRecords, Doctrine 2 is based on Data mapper pattern.
Both can do same things, but there are some significant differences between the two.
Generally speaking Data mapper is less "developer-friendly" but should have better performance. Why? Actually it is pretty simple. With active records each entity knows everything "around" itself, relation with other entities etc. With data mapper, entities are dumb and lightweight, there is a central entity (EntityManager/UnitOfWork in Doctrine2) which handles all the relation mapping. So in terms of memory usage and performance Data mapper should be faster.
The Doctrine guys say that Doctrine2 is a least 50% faster that Doctrine1 (there are other differences too, not just the design pattern).
If you feel up for it, you can even implement ActiveRecords over Doctrine2 data mapper. Look at this blog post. I'm using this approach just for the development phase, to keep as little code as possible. Once it gets into production I will kill the additional ActiveRecords layer, and rollback to the default data mapper of Doctrine2.
So the conclusion is that you can do everything with both, but in the same way you could say that you can do everything with raw SQL. If you are a beginner in the ORM world, I would suggest going with ActiveRecords, because it is simple and (usually) requires less code. On the other hand, if you are building a large, complex model, I think data mapper is the better option.
Maybe I got something wrong, but this is how I understood it.
As for the comparison between CodeIgniters ActiveRecords and Doctrine (1 or 2), I can't really say, because I never used CodeIgniter. One thing I am sure of, Doctrine has a lot more features than CodeIgniters default ORM. For example: result hydration, inheritance (single table, class table), prefetching, lazy loading, extra lazy loading, extensions, behaviours, optimization, proxies, datetime handling... It is a massive and full-fledged ORM with a lot of features, while my experience with any "default framework ORM" is that their main goal is to be simple as possible, so a newbie can get a hang of it very easily. Doctrine is a mighty beast, and for sure can do a lot of things in a more efficient and/or logically more correct way than the built in CodeIgniter ORM. The downside is, that it takes more time to learn and code, and it is a huge library, with thousands of files, so just to get everything running adds some overhead compared to a lighter alternative.