Season and frequency
First of all Slovakian Tatra is almost closed between November-June. Polish not at all. But be experienced, prepared and equipped for ice and snow conditions avalanches.
Polish Tatry is overcrowded in high season July-August. Slovakian part is as well but less.
I would say that in Slovakian Tatry (Zapadne- Western Tatry) the less frequented area is Rochace.
Basically weekends in Polish Tatra are not advisable for somebody who likes to be on its own.
Nizke Tatry is less crowded.
I believe that the crowded paradise can be also visited- if you are interested search for Slovensky Raj.
Less crowded places:
Chocske Verchy is also interesting.
Other ones are Mala and Velka Fatra.
The best time to come is September and October for my money due to low rainfall statistics and less tourist.
Polish/Slovakian/Czech hikers who prefers calmer places often go to Ukrainian or Romanian Karpaty , some also to Bulgarian or Serbian, Macedonian mountains.
But you will also find less frequented mountains in countries mentioned by you.
In case of emergency (read often as frivolously) one pays on his/her own for rescue in Slovakia. I pay in such case for somebody else in Poland (from my taxes)so far. So you may need insurance which cover such accidents in Slovakia.
Slovakia has excellent of system train in Tatry area (electricka).
Minibuses system on Polish side is pretty efficient.
Buses are the best for longer routes in both countries.
You can easily depend on inns, huts etc.
Nizke Tatry- you can make a ridge trip including usage of standard mountain hostels in area of Chopok and self-service (and crowded in the high season) huts such as Andrejcova and Ramze.
Tatry- you can make trips from one mountain hostel to another. It is sometimes better to stay in one place longer and make more demanding routes 'on light'.
Some prefer stay few nights in valley's villages.
Many Slovakian mountain accommodation places have rather status of few stars hotels than hostels, which means they will not allow you to stay if no places.
In many Polish mountain hostels you are allowed to sleep on the floor if no space.
Generally if you want to have a better comfort booking in advance could be an option.
Apart from the crowd, some places in Polish Tatry are quite distinctive like:
the most difficult route Orla perc (The Eagles' Path), Morskie Oko- The Sea Eye Lake (probably the most frequented place in all Tatry, a bit overrated for me), dolina (Polskich) Pieciu Stawow- Five (Polish) Lakes Valley is stunning. Mylna (Confusing) cave is not long but only one to visit on your own in Tatry.
Length of stay
It is easy to imagine the someone who likes not only hiking, but also wants to see (especially in a rainy day ) historical/folklore spots, guided caves, UNESCO heritage places such as wooden churches and thermal pools (not my favourite), can spend in this area not only two, but even more weeks.
Krakow is only ~~2,5 hours from Zakopane by bus. The distance is 120km but we complain our roads.