I have a number of clients who use CiviCRM on a subdomain separately from their main content site. It has both pros and cons.
The main issue addresssed here is the risk of security lapses on the main site exposing CiviCRM. The reason a separate site can be more secure is that you will typically have more modules and functionality on your main site where ease of use is the primary goal, whereas with CiviCRM, security is a much higher priority. Keeping the CiviCRM site with fewer 'ease of use' tools and generally fewer functions will almost always make it more secure, at least in principle.
Analogously - you can also provide hassle-free administrative access to designers and content editors without risk of inadvertently exposing CiviCRM.
The primary use case of your main site will typically be anonymous access by visitors, whereas most of your access to CiviCRM will be authenticated. The kind of server setup for those two types of access are frequently at odds, so having them on separate servers can make for better use of machine resources. e.g. you can put your main site on a relatively inexpensive hosting service with heavy caching for anonymous visitors.
Keeping internal resources to understand how multiple systems works can be a hassle and/or expensive. This is really the key driver of the decision - if you already have reasons to keep your CMS where it is, or to move it to a different CMS than your CiviCRM is on, then a split install is often a good choice.
The main other cost of the separate installs is maintenance - keeping up two sites instead of one, and updating design elements to keep the visual link between them convincing.
3. Server costs.
Usually, two sites is going to cost more than one, but see note 2. above. Specifically - by having the lower-security functionality all together, you can use your budget more effectively, e.g. you can have a managed CiviCRM install and use more off-the-shelf (read cheap/do-it-yourself) tools for your CMS.
I would also note that you don't need fancy machine-level integration as suggested by ErikH - you just need links back and forth, and an internal clarity that the two pieces are only loosely joined.
For clients who are migrating from other solutions (e.g. separate online processsor services, mailing lists and event services), putting all these together on one CiviCRM site that they manage is already a huge step forward in integrating those pieces, and having that functionality relatively loosely linked to the content site is not usually a show stopper, and in some cases a desirable state.
Here are few sites that are hosting CiviCRM on a separate subsite: