API designers deprecate classes and methods as a signal that they shouldn't be used. Usually because there's a better way to do the same thing, but occasionally (as in the case of Java's
Date constructors) because the existing methods have unintended consequences. Deprecation is the first step in controlled breakage of backwards compatibility, a warning for users that they can't rely on the functionality always being present.
Of course, in the real world, deprecated functionality rarely gets removed. The
Date constructors have been with us since Java 1.0.x and show no sign of going away. Indeed, of all the major frameworks that I've used, the only one that takes deprecation seriously is Spring — and even then, only minor things seem to get removed; you can still compile a program written for Spring 1.x with only a few changes.
However, leaving deprecated code in your codebase is a form of technical debt. You run the risk that the functionality will one day be removed, and that you'll have to scramble to replace it. But more important, if your code contains a mix of deprecated and non-deprecated features, you have to expend extra effort to maintain it.
The worst case that I've seen was a company that built websites for clients. One of their features was a content management system: a tool that allowed the client to create their own content, inserted into the page when various conditions were met. Standard stuff, but this company actually had three incompatible content management systems, all home-grown. Each was developed to replace the previous.
The reason that all three systems remained in active use was that there was a cost to convert to the newer framework, and neither the company nor their clients were willing to pay that cost. In effect, they chose to leave deprecated code in their codebase.
Since the frameworks were all built in-house, there was no reason to worry about them going away unexpectedly. But what did go away were the developers who had built the frameworks, and with them the knowledge of how the frameworks worked. Although the older frameworks didn't have a lot of bugs, when one did appear it was a crisis situation, with a mad search to find someone with enough knowledge to track it down and fix it. Even without bugs, the lack of knowledge meant that every enhancement to the client site took extra time, as each new developer learned enough to avoid breaking the CMS.
This situation continued for as long as I worked at that company; for all I know, they've added a fourth or fifth system to the mix. Although the old systems imposed an unnecessary cost on each project, that incremental cost was always lower than the cost to upgrade. But over time, the costs added up, and the client paid much more than an outright conversion (which the company didn't mind, as that increased revenue).
Worse, developers did all they could to avoid contact with the older systems, or to have their names on any checkins associated with it. Getting pulled onto a CMS problem meant that your own project schedules would slip, and you'd have to listen to upset project managers waste more of your time. Much easier to let the other team sink or swim on their own.
I'm not saying that ignoring deprecations will create disfunction and discord within your development team. But do you want to risk it?