I haven't ever used the Casio CTK-7200 or Roland BK-3, but I do own a Yamaha PSR-443, so I'll discuss it.
The PSR-E443 would certainly be a good step up from the PSR-E333, because it's similar enough to the PSR-E333 that you should feel comfortable with the upgrade, but it also has some features that the PSR-E333 lacks:
- PSR-E333 has no User files capability. -- PSR-E443 can save and load User files having 100 different file names. (User files are similar to Backup files, but can be loaded from a USB thumb drive plugged into the back of the keyboard, as opposed to being transferred to/from a computer via the Musicsoft Downloader software.)
- PSR-E333 has no Registrations. -- PSR-E443 has 32 Registrations. (Registrations let you save your keyboard settings for later recall.)
- PSR-E333 can "register" only 1 external Style at a time. -- PSR-E443 can "register" 5 external Styles at a time.
- PSR-E333 can save 5 User Songs at a time. -- PSR-E443 can save 10 User Songs at a time.
- PSR-E333 doesn't have a Pitch Bend wheel. -- PSR-E443 has a Pitch Bend Wheel.
- PSR-E333 has no Functions for modifying the Voices (but they can be modified through MIDI commands). -- PSR-E443 has Functions for modifying the Main and Dual Voices. ("Modifying" in this case refers to changing the Attack Time, Release Time, Filter Cutoff Frequency, and Filter Resonance settings of the Voices.)
- PSR-E333 has no "Live Control" knobs. -- PSR-E443 has 2 assignable "Live Control" knobs. (The "Live Control" knobs let you modify the Cutoff/Resonance, Reverb/Chorus, and Attack/Release settings while playing, without having to navigate through the Functions menu-- and they can also be mapped to various MIDI control messages in a DAW.)
- PSR-E333 has no "Aux In" port. -- PSR-E443 has an "Aux In" port. (The "Aux In" port lets you feed an audio signal-- e.g., an MP3 song or VST instrument sound-- into the keyboard so that it plays through the keyboard's speakers; and you can filter out selected frequency ranges from the incoming audio signal.)
- PSR-E333 has no Arpeggio feature. -- PSR-E443 has an Arpeggio feature with 150 different Arpeggio types.
- PSR-E333 has no Pattern feature. -- PSR-E443 has a Pattern feature with 20 different Patterns, each having 5 different variations, so it essentially has 100 Patterns in all. (A Pattern is somewhat similar to a Style but is geared more toward "DJ effects" as opposed to "auto accompaniments.")
Aside from these differences (which are significant), the PSR-E443 also has more built-in Voices and Styles than the PSR-E333. Thus, the PSR-E443 should definitely give you a lot more to play with than the PSR-E333. On the other hand, the PSR-E443 still has a polyphony of 32 notes-- the same as the PSR-E333.
As for the Casio CTK-7200, I'm personally interested in it because of the features it has that the Yamaha PSR-E443 lacks-- namely, sliders for modifying the organ drawbar settings, as well as the ability to create new Styles ("Rhythms"), plus the various DSP effects. The Casio also has better polyphony than the Yamaha (48 versus 32 notes).
I'm not at all familiar with the Roland BK-3, but from what I see on Roland's web site it appears to be very promising-- and it has compatibility with GM2 and GS voices, which the Yamaha PSR-E443 does not. Plus, it has 128-note polyphony, which is far more than either the Yamaha PSR-E443 or the Casio CTK-7200.
In my opinion, polyphony is more important for "live" playing with an auto accompaniment. If you consider that you have only 10 fingers, the most notes you can play at a time are 20 (assuming each finger is pressing 2 adjacent notes, which isn't usually done unless you have "butter fingers")-- although if the notes are sustained after you raise your fingers from the keys, then some of the polyphony will be used for the sustained notes. Also, if you're layering two sounds together, each finger press will use 2 notes of polyphony-- plus, even with a single-layer sound, some sounds may be constructed such that each note uses up twice as much polyphony. Hence, the Yamaha PSR-E443's 32-note polyphony could conceivably translate into just 16 or even 8 notes played at once, depending on whether you're using layered ("Dual") Voices and/or Voices that use twice as many "sound elements" as normal. So if you're going to be using the keyboard for recording to a DAW, the polyphony might not be a big deal (after all, most classic analog synthesizers were monophonic!)-- but if you want to use auto accompaniments, or play back a multi-track MIDI recording through your keyboard, then the polyphony could be extremely important. So generally speaking, a keyboard with 128-note polyphony is going to be far superior than a keyboard with only 32-note polyphony. But even so, if you're recording tracks one at a time using a DAW, 32-note polyphony may be more than adequate.
What you really need to do is try out each prospective keyboard at a music store to see how it sounds and how it feels, so you can get a better sense of whether it will be suitable for your purposes.