So, Haydn with 106 symphonies and Mozart with about 50. Mozart put an awful lot of his energies into opera and concertos where probably the core of Haydn's work over the years was with the symphony. Plus, Haydn tends to be consistently underrated these days. So, for those reasons I am going to pick six symphonies by Haydn and four by Mozart. And, to avoid pointless controversy over ranking, I am going to list them in chronological order. The four by Mozart are pretty easy, but picking just six out of the enormous Haydn output, that's the tricky part. so here goes.
1. Symphony No. 22 in E flat major, "The Philosopher" (1764) by Joseph Haydn. This is one of the best early symphonies, written after he had joined the Esterházy household. The instrumentation is very unusual as it includes two English horns as well as two horns and strings. This is the only symphony in history to be written for this combination. It is also unusual as it follows the old-fashioned slow-fast-slow-fast layout of movements. The origin of the nickname, not given it by Haydn, is unclear, though some speculate it is because of the dialogue between the horns and English horns in the first movement. The horns are the curled hunting horns, but English horns are neither English, nor horns as they are a kind of tenor oboe.
2. Symphony No. 45 in F# minor, "Farewell" (1772) by Joseph Haydn. One of the most popular and original of Haydn's symphonies. It ends with a unique bit of theatre: during the last movement the musicians one by one pack up and leave, blowing out the candles on their stands as they go. Finally there are only two violins left to end the movement. The reason for this was that the Prince had kept his household at their country estate long past the usual date. At the end of summer everyone returned to Vienna for the winter and the musicians were able to rejoin their families. Haydn ended the symphony this way as a gentle reminder to the Prince that everyone wanted to return home.
3. Symphony No. 60 in C major, "Il Distratto" (1774) by Joseph Haydn. This symphony is derived from incidental music written for the play "Le distrait" by Jean-François Regnard. For this reason it not only has six movements but has a very unusual character with quick mood-swings and an almost absurdist and discordant flourish in the last movement. Conductor Kenneth Woods describes it as "Haydn at his absolute boldest."
4. Symphony No. 29 in A major, K. 201/186a (1774) by W. A. Mozart. This is my favorite of Mozart's early symphonies, written when he was eighteen years old. The opening theme of the first movement is one of those great themes that is very hard to get out of your head!
5. Symphony No. 36 in C major, "Linz" K. 425 (1783) by W. A. Mozart. This symphony was written in a mere four days while visiting the town of Linz on his way back to Vienna.
6. Symphony No. 82 in C major, "The Bear" (1786) by Joseph Haydn. This is one of the six "Paris" symphonies commissioned by the Concert de la Loge Olympique for their concert series and represents the beginnings of Haydn's international fame. The nickname, again not chosen by Haydn, comes from the theme of the last movement whose bagpipe-like drone was similar to those used to accompany dancing bears.
7. Symphony No. 88 in G major (1787) by Joseph Haydn. This symphony is very popular despite being part of neither the set of "Paris" symphonies nor the set of "London" symphonies and not possessing a distinctive nickname! What it does possess is possibly the most cheerful and exuberant finale ever written.
8. Symphony No. 40 in G minor, K. 550 (1788) by W. A. Mozart. Written, along with Nos 39 and 41, in just a few short weeks in the summer, this is one of his last symphonies. A number of features, such as beginning the first movement with an accompaniment figure, were very influential with later composers. One of just two symphonies written in minor keys by Mozart, this is much darker in mood than is typical for symphonies in the 18th century.
9. Symphony No. 41 in C major, K. 551, "Jupiter" (1788) by W. A. Mozart. The companion to No. 40 and Mozart's last symphony. The last movement is one of the most stunning accomplishments in the genre. A heaven-storming finale it is a fugato with five different themes and was possibly the inspiration for Woody Allen's remark that Mozart is the proof of the existence of God.
10. Symphony No. 103 in E flat major, "Drumroll" (1795). Written in and for performance in London on the second of Haydn's triumphant tours there, this is his penultimate symphony and unique for beginning with a tympani roll. All his late symphonies are superb and I could have chosen any one of them, but I just happen to really like this one.