Dennis,
they're two quite similar cartridges in many respects apart from a little matter of 100 years age between them. Both are long-necked and with COALs >3 inches need a long action, although you can get away with many shorter single-shot actions for target shooting if you're willing to unload a live round by removing the bolt.
The big differences are case capacity - ~57gn water for the '55 and ~66 for the 284. That's a 16% case volume increase and the rule of thumb is that all other things being equal (PMax and barrel length) divide the % increase in case capacity by four to get the likely % MV increase, ie ~4% or 120 fps for a 140gn class bullet if the '55 is pushing it out at 2,900 fps. And sure enough, the 6.5-284 has no trouble getting bullets out at over 3,000 fps, but if you look at the cartridge guide elswhere on this site, most top US competition users weren't pushing it as hard as that.
In practice, things aren't quite equal, at least not officially. CIP PMax for the '55 is only ~55,000 psi in deference to the number of old rifles chambered for it, whilst the CIP sets 59,400 odd psi for the 6.5-284 Norma, actually a low figure for this vintage of cartridge when 65,000 psi maxima aren't too uncomon. In practice, handloaders using good brass in a modern action undoubtedly load both to rather higher pressures. With Lapua brass in a modern rifle, there's no reason not to run the '55 at getting on for 60,000 psi. Both have good quality brass available, Norma and Lapua.
Both can provide superb precision in a good rifle and barrel and make superb 1,000 yard performers, but lose out equally to the .284 Win, 7mm Shehane and 7mm short mags when the wind becomes 'tricky', not to mention Steve Blair of this forum and his soulmates with their .300WSMs and 215 or 230gn Hybrids!
In practice, the '55 has a slightly too small capacity case, and the -284 a bit too large. Some are of the opinion that the '55AI 40-deg is the optimum 6.5 match cartridge falling between the two. I seriously considered this option at one time, but the main (really only) downside is dies, as a custom set is the norm. (A common trick is to start with Redding Comp 6.5X55 dies and machine the AI case-form into the relatively soft case sleeve insert.)
The other side of the coin is that while the '55 gives 100-150 fps less MV than the -284 version, it gains in barrel life. Both use similar powders - 4831s, N560, N165, Re22 and similar. Both are usually easy to tune.
The other thing to note is that the '55 has a slightly larger dia. case-head than the originally Mauser norm of 0.473" - some 0.010" or so. Many 308 face bolts accept the larger case without alteration, but a minor bit of relieving is sometimes necessary.