Sweden has won more by most counts. /u/vonadler did a good summary of all the wars fought between Swedem and Denmark in this thread.

I'll copy my old post and add victories and draws.

It depends a bit on wether or not you count the various fighting within the Kalmar Union as Dano-Swedish wars or not.

I'll count Sweden as existing from the crowning of Knut Eriksson (Erikska) as King 1173. This is the first time we can confirm properly that someone was King over Östergötland, Västergötland and Svealand at the same time, and the three Kingdoms did not splinter afterwards.

I'll do a list and a conservative and a generous estimate.

1205-1210: The Danes support Sverker Karlsson (Sverkerska) in his fight with Erik Knutsson (Erikska) over the Swedish throne. Sources indicate Danish troops were directly involved in the Battle of Gestilren 1210, so I'll count this as one for the generous estimate. The Danish-supported side lost, so I'll count this for Sweden.

1360-1361: The Danish King Valdemar Atterdag captures Skåne, Halland and Blekinge (purchased by Sweden earlier) and Gotland (mostly an independent Peasants' Republic, but nominally belonging to the Swedish crown). This will count for both the generous and the conservative estimate. The Danes clearly won here, as Valdemar occupied a lot of terrain.

1389-1398: King Albrekt of Sweden and his supporters are defeated by the forces of the nobility and Queen Margareta of Denmark and Norway and the Kalmar Union comes into existance. I'll count this one for both the generous and the conservative estimate. Danish-supported forces won here, so I'll count this one for Denmark.

1434-1436: A revolt led by Engelbrekt Engelbrektsson leads to the first estates parliament in Sweden to declare King Erik of Denmark, Norway and Sweden deposed as King of Sweden. I'll count this as one for the generous count as it is a revolt within the Kalmar Union. The rebellion ousted the Danish King, so I'll count this one for Sweden.

1448-1450: As King Kristoffer dies suddenlly, the estates parliament elect the Swedish prominent nobleman Karl Knutsson (Bonde) King of Sweden. Both he and King Kristian I of Denmark are elected King of Norway by their respective supporters and war breaks out over Norway. Karl Knutsson (Bonde) quickly loses, but it is a real war. I'll count this one for both the generous and the conservative estimate. The Swedish side lost the quest for the Norwegian throne, so I'll count this one for Denmark.

1452-1457: War between King Karl Knutsson (Bonde) and King Kristian of Denmark (with a cease fire 1453-1455). In the end, Karl Knutsson (Bonde) is deposed and Kristian is elected King of Sweden too. I'll count this one for both the generous and the conservative estimate. The Danish side won this one.

1463-1471: Since King Karl Knutsson (Bonde) had returned as King, new fighting broke out as King Kristian tried to enforce his claims to the Swedish throne. Karl Knutsson (Bonde) died 1470, but his supporters rallied around Sten Gustavsson (sture) and decisively defeated the Danish army at Brunkeberg (nowadays inside Stockholm 1471). I'll count this one for both the generous and the conservative estimate. This one is a Swedish victory.

1501-1520: Sten Gustavsson (Sture), having been deposed by Swedish supporters of the Kalmar Union 1497 returns and revolts against King Hans of Denmark. Sten Gustavsson (Sture) assumes control of Sweden again and war continues until Sten Svantesson (Sture) is defeated by King Kristian II of Denmark at the Battle of Bogesund 1520. Sten Svantesson (Sture) is murdered and King Kristian II orchestrates Stockholm's bloodbath. I'll count this one for both the generous and the conservative estimate. This is a Danish victory.

1521-1523: Gustav Eriksson (Wasa) leads a rising against King Kristian II and finally ends the Kalmar Union. As part of the fighting happens when Gustav Eriksson (Wasa) has been elected King of Sweden, I'll count this one for both the generous and the conservative estimate. A Swedish victory. From here you can say that Sweden exists as a nation-state.

1534-1536: Sweden intervenes in the Counts' feud in Denmark to ensure that King Kristian II can not regain the throne in Denmark. As it is an intervention in a civll war, I'll only count this for the generous estimate. While the intervention keeps Denmark from meddling in Swedish affairs, it does not gain Sweden anything from the fighting, so this is a draw.

1563-1570: The Nordic Seven Years' War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. This one is mostly a draw as no territory was exchanged. Sweden had to pay 150 000 thalers to gain Älvsborg back, but the Danish King had to abstain his claim on the Swedish throne. I'll count it as a draw.

1611-1613: The Kalmar War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. No territory was exchanged, but Sweden had to pay a monstrous war indemnity to gain Älvsborg back (again) of 1 000 000 thalers, so I'll count this as a Danish victory.

1643-1645: Torstensson's War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. Sweden gainst territory, so it is a Swedish victory.

1657-1658: Karl X Gustav's War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. Sweden gainst huge amounts of territory, so I'll count it as a Swedish victory.

1658-1660: Karl X Gustav's second War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. Denmark regains some of the territory lost in the previous war, so I'll count it as a Danish victory.

1675-1679: The Scanian War between Denmark and Sweden. Full on war. I'll count this one for both the generous and the conservative estimate. No territory exchanged, so I will count this as a draw.

1700: First Danish participation in the Great Nordic War. A very short war, where Swedish forces land outside Copenhagen and force Denmark to leave the anti-Swedish alliance. As it is short, and Denmark took part in the Great Nordic War later as well, I'll only count this for the generous estimate. No territory exchanged, so I'll call it a draw.

1709-1719: Second Danish participation in the Great Nordic War. Full on war. I'll count this one for both the generous and the conservative estimate. No territory exchanged, so I'll call it a draw.

1788-1790: The teather war. Sweden attack Russia, and Denmark honours its alliance with Russia and declares war. There's little actual warfare between Sweden and Denmark, though. I'll only count this for the generous estimate. Very little warfare and no territory exchanged, so I'll call it a draw.

1808-1809: The Danish war. Denmark, allied with Russia, which prodded by Napoleon attacks Sweden to force it into the continental system, declares war on Sweden. The fighting is limited, as the Danish forces in Norway lack supplies due to a blockade by the Royal Navy and the Danish, Spanish and French forces in Denmark cannot get across the Sound to invade Sweden. However, Sweden attempts to invade Norway but is turned back. I'll count this one for both the generous and the conservative estimate. No territory exchanged as part of this, so I'll call it a draw.

1814: The invasion of Norway. At the peace in Kiel 1814, Denmark had to give Norway to Sweden, but the Norwegians themselves tried to declare independence. As it is hard to say that it is a war against Denmark despite the Norwegian army fighting being trained, equipped and officered by the Danes, I'll count this only for the generous estimate. Sweden gains Norway, so I will count this one for Sweden.

1814 was the last war Sweden fought.

This gives us a total of;

Generous: 21 wars between Sweden and Denmark.

Conservative: 15 wars between Sweden and Denmark.

7 won by Sweden, 7 won by Denmark and 7 draws.