It'll be a definite run [of a file system check] if I'm booting after letting the laptop runs out of battery.
On every mount, Ext3 will simply replay the journal, eliminating the need to do a full file system check. But really, please avoid a hard shut down of the system! It's not a good practice if your data is valuable. Instaed, let Ubuntu just shut down your system gracefully on a low battery.
To see if your file system will be checked on the next boot, run
dumpe2fs -h /dev/sda1 | grep -iE "(mount count|check|state)"
If you really need to hard shut down without taking the time to going through the shut down procedure, then at least shut down your file systems cleanly before pulling the power - e.g. by doing Alt + sysrq + REISUB (but use O instead of B to halt).
Is it the case that the file system on my Linux partition (ext2/3) is more fragile than the file system on my windows partition (NTFS)?
No. I believe the Ext3 file system is very stable and very unlikely to break. The checks just make sure the structure is intact and that possibly corrupt blocks are being hidden (moved to
lost+found) and that the files will be in a known consistent state.
Is it the case that Windows checks for errors and solves them in the background?
It's not the right place to ask how Windows works, but I really prefer to know whether my file system can be recovered to a known state before my applications are allowed to start and write to the disk. Doing a check afterwards while applications are already writing, it can make things a lot worse.
Long story short: checking the file system before it goes "live" and writeable, is of your own benefit.
Is it the case that Windows doesn't check for errors, and Ubuntu is just more vigilant in terms of file system health?
As became clear later in the comments, you appear to be using Ext2 still. That version does not feature any form of a journal, as psusi points out. The above does only apply to Ext3/4 and most other journaled file systems. Please avoid the use of such and old file systems if your data is valuable in the conditions of unclean shut downs. There's no reason to use it still for regular Ubuntu desktop installations.
Newer generation file systems
Additionally, I wanted to point out that a newer generation of file systems is starting to become more and more mainstream. For Ubuntu/Linux this is Btrfs, a design similar to ZFS. It will be way more intelligent in fixing corruptions and is self-healing on-line. However, it will not be able to do magic on pulling the power from a running system - there's still risk in losing data you wrote just before you pulled the plug.