Hi Danielle, I have a few coments a little different to the other themes in this thread.
SHIFTING TO 2.X Firstly, there is pain in the upgrade to 2.X. You can end up with a lot of extra stuff in your database if you do the upgrade paths suggested. There are unresolved issues with what happens to legacy files. However you do get users/old data shifted across.
If you go another way, setting up a nice clean shiny new 2.4 server and importing courses one by one you are up for a lot of time. You can't shift users, but this can be managed. You can hit the "big file" limit. https://tracker.moodle.org/browse/MDL-34388 You can have audit/history requirements from your institution that mean this is not possible.
Many places have run two systems at the same time, and had a slow motion change over. This is my favoured route.
With either approach, you have the problem of plugins and customisations. You have user training needed (like the new file handling). And you DO need an upgrades system in terms of power and versions of PHP etc.
SECURITY. I'm not sure of Marc's comment about secturity: I'd need more information and specifics, but sometimes, unpopular as his posts are, he is at least partly right.
Delaying a shift may one day lead you to the place where your servers have moved on, a new version of PHP doesn't support 1.7 etc etc. But this is purely speculation. I remember having to migrate (rewrite) my DOS database programs to WINDOZE. 
---
This is where you may not like my comments. You say: "Our school currently uses Moodle 1.9.7 and are happy with it".
This may well be a completely true statement. But trying to find an analogy: "We have an open fire, and we are quite happy with it. We don't need air conditioning".
There are a lot of nice new things in 2.4 core. Drag and drop. Duplicate activities. Section descriptions. Insert pictures in student posts. Assignment improvements. Conditional release. Better editor. New theme engines. Responsive design. Quiz improvements.
Some things are improved, but not fixed fully yet: single section display. (ie the scroll of death is partly solved). Sharing files between courses. (ie the silo of the course it partly solved). Some improvements to core enrolment plugins/functionality (but still no automation of user management and enrolments, so there is a huge overhead in managing accounts unless you have provisioning done another way). Navigation. (We now sometimes have a Navigation SOD)
Some really basic things things remain unresolved in Core Moodle. A small number of things have gone backwards.
---
However, You may not really be using Moodle 1.9.7 core, but have all sorts of addons and hacks, so you really don't NEED to move. I met someone years back who I respected. They said "We use Moodle and it is great" For me at the time there were huge show stoppers in Moodle. Then I found out that this guy did not realise they had a system with Moodle + hacks, + plugins and + customisations to make things easy to work. No scroll of death, no user management overheads, an alternative forum, provisioning courses with activities with less clicks, custom reports etc. So we were not talking about the same entity. This is the problem of maintainability if you stick with a highly patched 1.9.x. A faith statement: I believe that most of these issues (but possiblynot all) can be worked on with careful choice of plugins in a 2.4 install.
In summary:
- probably you will (sooner or later) find some cogent arguements to reach a conclusion you MUST ugrade.
- It may not be security, it may just be the problem of an old system, it may be a need to upgrade something, it may be a need for a new functionality.
- There is PAIN in the ugrade.
- But there is a lot of nice stuff in the new core Moodle that users will like.
- And with few basic plugins from the plugin directory you can make up for some of the deficiencies of Moodle.
- You may find a whole new world of good things.
Personally, I think that if we are on 1.9.x we need to consider how to make the shift from 1.x to 2 and when (using an approach with the least pain), and move beyond the question "if" we make the shift. While I still find some things difficult in 2.4, on balance we have gained far more than we have lost, and for other reasons, there may be problems eventually if we don't make the shift.
Much more than I ment to write.
-Derek