Context clues point to you getting healing and curing
'Also' is used in both ways in the PHB
When one ability has multiple effects, ideally the text would tell us whether these effects are mutually exclusive (through language like 'instead') or are complementary (through language like 'in addition'), and it often does.
For example, the two uses of Lay on Hands are described as being alternates of one another1, while the two features of the Elemental Adept Feat are explicitly co-occurring2.
Sometimes, however, the rules text uses 'also', and such uses can be grammatically ambiguous. This is the case with Physician's Touch. At third level, you can use use Hand of Healing to restore hp. It could be that at 6th level, you can also use Hand to cure disease; you now have two different things you can use Hand for (also = alternatively, instead of). Or, it could be that at 6th level you can use Hand to both restore hp and also cure disease at the same time (also = in addition to). 'Also' can be used either way.
Within the PHB, we can find both uses of also. For example, in the section on using oil3 and in the clone spell4, 'also' means an alternative in a list of things. You can throw oil as an attack and you can also pour it on the floor, but you can't do both with the same flask. You can have a clone grow to maturity in 120 days or you can also have it grow to a stage younger than maturity - but you can't do both with the same clone. Here 'also' indicates that these are alternative choices.
On the other hand, sometimes 'also' means in addition, rather than as an alternative, as is the case with dominate person5, where you can use your action to control the action of the target, and in addition, you can also use your reaction to control its reaction.
So 'also' can be, and is, used in both ways in the core rules. I've chosen specific examples as cases in which the context makes it clear what the meaning of 'also' is. However, that clarity is not present in the case of Physician's Touch. Both meanings of 'also' are possible, which is what makes it particularly difficult to understand what the rule intent is for Physician's Touch - hence the question.
But 'when' is a clue that here, 'also' means additionally
Fortunately there is one context clue present in Physician's Touch, in the form of:
When you use Hand of Healing
In every example of using 'also' that I could find in the PHB, the use of 'when' signaled that 'also' was being used in the sense of 'additionally', rather than in the sense of 'alternately'. That is, some base effect applies, but you can add another effect under certain circumstances, and those circumstances are signaled by the use of 'when'. This can be seen in, for example, arcane lock6 where you and designated creatures can always open the lock; this is the base condition, but in addition to these designated creatures, you can 'also' have someone not designated open the lock, when they speak the password. Similar language is used to similar effect in guards and wards and symbol. The Share Spells feature of the beast master ranger7 has 'additional also', targeting you and your companion 'when' you cast (not 'alternate also' targeting one or the other). Similar language is used in find greater steed.
Not every 'additional also' uses the 'when' language, as we saw with dominate person. But in cases where 'when' is used, it is a clear indication that the intent is 'additional also'. Knowing this, we can interpret the ambiguous Physician's Touch as an 'additional also'. "When you use Hand of Healing" is the base condition - you are already restoring hp. The Physician's Touch permits you to remove disease in addition to the healing.
1 Lay on Hands (emphases mine):
As an action, you can touch a creature and draw power from the pool to restore a number of hit points to that creature, up to the maximum amount remaining in your pool.
Alternatively, you can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it. You can cure multiple diseases and neutralize multiple poisons with a single use of Lay on Hands, expending hit points separately for each one.
2 Elemental Adept Feat (emphasis mine):
Spells you cast ignore resistance to damage of the chosen type. In addition, when you roll damage for a spell you cast that deals damage of that type, you can treat any 1 on a damage die as a 2.
3 Using oil (emphasis mine):
As an action, you can splash the oil in this flask onto a creature within 5 feet of you or throw it up to 20 feet, shattering it on impact...You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level.
4 Clone (emphasis mine):
This clone forms inside the vessel used in the spell’s casting and grows to full size and maturity after 120 days; you can also choose to have the clone be a younger version of the same creature.
5 Dominate person (emphasis mine):
You can use your action to take total and precise control of the target. Until the end of your next turn, the creature takes only the actions you choose, and doesn't do anything that you don't allow it to do. During this time you can also cause the creature to use a reaction, but this requires you to use your own reaction as well.
6 Arcane lock(emphases mine):
You and the creatures you designate when you cast this spell can open the object normally. You can also set a password that, when spoken within 5 feet of the object, suppresses this spell for 1 minute.
7 Beast Master Ranger (emphases mine):
Beginning at 15th level, when you cast a spell targeting yourself, you can also affect your beast companion with the spell if the beast is within 30 feet of you.
Your spells normally function on their target; that is the base condition. When you target yourself, the spells can affect your companion in addition to the default target.