What are the differences beetween European (230 V 50 Hz here in Finland) high pressure sodium or metal halide and mercury ballasts? 

These lamps have several types of exact same wattage, so why they have different ballasts then?

Are there significant differences in arc voltage or do lamp ignition have some special needs of the ballast?

For a given wattage there's a large number of possible voltages and currents.

HPS lamps usually have a lower voltage drop than mercury lamps, so to get the same wattage, more current is needed.

And indeed many types of HID lamps need an ignitor while mercury lamps don't.

Exactly. Different lamp types have different electrical characteristics, it's not just about wattage. A 250W MV lamp is very different from a 250W HPS or MH.

There are MH lamps that designed for MV ballasts (Like Philips HPI-T) and ones that designed for HPS ballasts (Osram HQI-BT 400W/D).

It is common for some people to mistakenly interchange both types of same wattage metal halide lamps, which can sometimes lead to underdriven lamps and overheated ballasts.

The question was not that much if they are different, but the reasons why they are different.
Speaking about 230V areas, where pretty much all ballasts are a series choke.

What to take into account when designing electrical specs for a certain desired lamp power rating for a new lamp type:
- Isn't something suitable already standardized in use for some older technology? If so, use that specs for that new lamp and design the new lamp parameters for it. That will allow to construct the gear from already established, so cheaper parts, design just the few extra parts that are not available yet (e.g. an ignitor,...). That way it will be easier and cheaper for your customers to adopt to your new lamp technology. Of course, this option was not available for the early HIDs, but definitely was the way how e.g. all the European MH's got their specs: They were intentionally designed to suit the then existing HPS or MV ballasts.

- The lower the current, the lower drop the ballast will have, both reducing the VA the choke has to handle, so the cheaper and more efficient that ballast would be. The lower current means higher arc voltage.
- In order to let the ballast reliably operate the arc, the arc voltage can not go much above 50% of the mains voltage. With larger arcs (higher power) the arc is generally more stable so it allows this percentage to be small bit higher, but only a bit. That is limitting the maximum arc voltage the lamp may have on a given minimum mains voltage. Lower wattages end up at around 100V arc, higher wattages up to 130V.
- Many lamp chemistries tend to rise their arc voltage as they age, so the lamp specs must be designed so to stay within the arc voltage limit (see above) over the whole lamp life.

So when taking the stable arc voltage nature of an MV, the arc voltage could be directly designed at the 50% mains maximum the series inductor ballast allows. So MV specs end up in the 100..140V range (corresponding to the 50..1000W power range in use at 230V mains). The current ratings then came as the result of getting to the desired power rating with that arc voltage
The HPS tends to increase the arc voltage as it is aging, so to get reasonable life, 30% margin must be left for the new lamp specs. So the specs for new HPS (= their "nominal" rating figures) end up with 70..100V arc, which is then anticipated to become 100..140V at the anticipated EOL. As the market asked for the same nominal power ratings as was common with MV, HPS ended up requiring dedicated ballast line.
And as mentioned before, when MHs came to widespread use (mainly the pulse start ones; I'm talking about European 230V market), both MV and HPS standards were already established and mainly in mainstream use (the MH was rather a high color quality niche), so allowed the lamps to be designed to match one of them. So most wattages ended up to be specified along the HPS, some higher power ones (where achieving the few extra percent on efficacy figure pays more in energy savings and where the designs were more stable) were optimized for MV specs. Although the optimum for MH may have been somewhere between MV and HPS specs (it drifts the arc voltage, but not as much as HPS), the extra percent or so in efficacy was not enough to warrant dedicated ballasts for this rather niche family.

In the 120V area the MV specs were just taken over from Europe, as it needed a transformer on the 120V anyway, the MH were originally designed tofit MV ballasts, although later it showed up the reliability was not that great, so the OCV spec was revised. But the elevated OCV still maintained MV compatibility, so it was seen as not that big deal.
The HPS needed different ballasts anyway and except lower wattages it was necessary to be a transformer anyway, so lamp specs were designed to just optimize efficacy at each wattage (just the lower wattages were specified to suit simpler series choke, so with 50V arcs), so each power level end up with its own arc voltage (that would need a transformer even at 230V, so for 230V the HPS were respecified to stay with a series choke).
And pulse MH, because they came from 230V Europe, retained their original specs. As these specs were mostly based on European HPS, it meant dedicated ballast lines for the 120V markets.
So at the end unlike the 230V world, where each wattage uses just one of two ballas specs for virtually any lamp chemistry so many fixtures may run multiplr lamp types (usually HPS and pulse-MH) of the same wattage, in the "120V" markets pretty every lamp type needs dedicated ballast.

It is common for some people to mistakenly interchange both types of same wattage metal halide lamps, which can sometimes lead to underdriven lamps and overheated ballasts.

I am not surprised by that at all. The information available about these lamps is so scarce. Even most datasheets does not tell clearly what type of gear these lamps need. And then there are ton of manufacturer-specific abbreviations of these lamps.

For example I have one quite new fixture which wants lamp type SAP-TS, and I have no idea what it is, and Google does not know also. I use Philips CDM-TD 70 W in this fixture, but it does not start easily.

That's where the American ANSI system scores. You have a lamp marked H38, you use a ballast marked H38, simple.

And thaks for other answers!

That's where the American ANSI system scores. You have a lamp marked H38, you use a ballast marked H38, simple.

For the situation where practically every lamp type needs its ballast and where ballasts are distributed as complete set packages (ballast transformer, ignitor, capacitor,...) it is indeed the practical way.

However when the ballast gear is assembled by the sparkie himself from components distributed separately and when one component part number is used for multiple lamp types but it is the component combination (e.g. a 70W HPS/MH ballast coil in combination with a "35..70W HPS" ignitor makes the assembly compatible with only HPS lamps,...) it could be misleading the same way as the present "lamp power+family" listing, it would add just another conversion table to use.

That's true, but all the US ballasts I have came complete with the necessary components where needed.

That's where the American ANSI system scores. You have a lamp marked H38, you use a ballast marked H38, simple.

That sounds to be very good system.

For the situation where practically every lamp type needs its ballast and where ballasts are distributed as complete set packages (ballast transformer, ignitor, capacitor,...) it is indeed the practical way.

However when the ballast gear is assembled by the sparkie himself from components distributed separately and when one component part number is used for multiple lamp types but it is the component combination (e.g. a 70W HPS/MH ballast coil in combination with a "35..70W HPS" ignitor makes the assembly compatible with only HPS lamps,...) it could be misleading the same way as the present "lamp power+family" listing, it would add just another conversion table to use.

I would still appreciate it much if for example HPS fixtures has standardized lamp type mentioned in type plate instead of some manufacturer specific abbreviation.